Geometry

Welcome to geometry. Before we can prove anything or measure anything, we need a shared vocabulary, and that vocabulary starts with three undefined terms that we describe rather than define: the point, the line, and the plane. Mathematicians leave these undefined on purpose. If we tried to define a point using simpler words, those words would need their own definitions, and so on forever. Instead we agree on a clear mental picture and build everything else from there. Do not let that worry you; by the end of this lesson these ideas will feel completely natural.

The three undefined terms

A point has no size at all, only a location. On paper we draw a dot, but a true point is infinitely small. We name a point with a single capital letter, such as point A. A line is a straight path of points that continues forever in both directions and has no thickness. We name a line with any two of its points and a small double-headed arrow above them, like line AB, or with a single lowercase letter such as line m. A plane is a perfectly flat surface that extends forever in every direction and has no thickness. A tabletop or a wall is a small piece of a plane, but the real plane never ends. We name a plane with a single capital letter or by naming three of its points that do not all lie on one line, such as plane ABC.

Parts of a line: segments and rays

From a line we carve two very useful figures. A segment is the part of a line between two endpoints, written AB with a small bar above the letters, and unlike a full line it has a definite, measurable length. A ray starts at one endpoint and continues forever in exactly one direction, written ray AB, where the first letter A is always the starting endpoint and the second letter shows which way it heads. Order matters for a ray: ray AB and ray BA point in opposite directions. Two rays that share the same endpoint and point in exactly opposite directions form a straight line and are called opposite rays.

Relationships: collinear and coplanar

Points that lie on one single line are collinear. Any two points are automatically collinear, because you can always draw a line through two points, but three or more points are collinear only if one straight line passes through all of them. Points or figures that lie in one single plane are coplanar. Any three points are always coplanar, but a fourth point may or may not join them in the same plane. Two more building-block ideas: an intersection is the set of points two figures share. Two different lines intersect in exactly one point, and two different planes intersect in exactly one line.

Measuring segments and the Segment Addition Postulate

On a number line the length of a segment, also called its distance, is the positive difference of the coordinates of its endpoints. We take the positive difference because length is never negative. For the picture above, AB = 9 - 2 = 7. The Segment Addition Postulate says that if point M lies between A and B on the segment, then AM + MB = AB. In the picture, AM = 5 - 2 = 3 and MB = 9 - 5 = 4, and indeed 3 + 4 = 7, which matches the whole length. This postulate is the segment version of a simple idea: the parts add up to the whole.

Worked example 1: a straightforward length

Point M is between A and B, with AM = 12 and AB = 20. Find MB.

Step 1: write the postulate. AM + MB = AB. Step 2: substitute the known values. 12 + MB = 20. Step 3: subtract 12 from both sides. MB = 20 - 12 = 8. Check: 12 + 8 = 20, which is the whole, so the answer MB = 8 is correct.

Worked example 2: a midpoint with algebra

A point that divides a segment into two equal halves is called the midpoint; at a midpoint AM = MB, so each half is exactly half the whole. Suppose M is the midpoint of segment AB, with AM = 3x + 1 and MB = 5x - 9. Find x and the length of AB.

Step 1: because M is the midpoint, the two halves are equal, so 3x + 1 = 5x - 9. Step 2: subtract 3x from both sides to get 1 = 2x - 9. Step 3: add 9 to both sides to get 10 = 2x. Step 4: divide by 2 to get x = 5. Step 5: find each half by substituting. AM = 3(5) + 1 = 16 and MB = 5(5) - 9 = 16, which agree, so the work checks. Step 6: the whole segment is AB = AM + MB = 16 + 16 = 32.

Common misconceptions

• Thinking a line is short. A line has no endpoints and never stops. What you draw on paper is only a piece of it.
• Confusing a ray's direction. Ray AB starts at A; ray BA starts at B. They are not the same ray.
• Assuming any three points are collinear. Three points are collinear only if one line passes through all three; often they form a triangle instead.
• Using a negative length. Distance is always the positive difference of coordinates, so it can never be negative.
• Forgetting the "between" condition. The Segment Addition Postulate only applies when the middle point truly lies on the segment between the two endpoints.

Recap

Point, line, and plane are the undefined terms we build everything from. A segment has two endpoints and a length; a ray has one endpoint and continues forever one way. Collinear points share a line; coplanar points share a plane. Length on a number line is the positive difference of coordinates, and the Segment Addition Postulate, AM + MB = AB, lets you find an unknown part when a point lies between two others. A midpoint splits a segment into two equal halves.

Sources

1. OpenStax, Prealgebra 2e, Chapter 9 (Geometry: points, lines, and planes). Available free at openstax.org.
2. Khan Academy, "Geometry: Lines" unit (points, lines, line segments, and rays). khanacademy.org.
3. Euclid, Elements, Book I, Definitions 1 through 4 and Postulate 2 (a point, a line, and extending a line). Clark University online edition.

An angle is formed by two rays that share a common endpoint called the vertex. The two rays are the sides of the angle. We measure the amount of turn between the sides in degrees, symbol °, on a scale from 0 up to 360. Angles are named three ways: by their vertex alone (angle B), by a number placed inside them (angle 1), or by three points (angle ABC), where the middle letter is always the vertex. We write the measure of the angle as m<ABC. Using three points is safest, because when several angles share a vertex the single-letter name would be ambiguous.

Classifying angles by measure

Angles are sorted into four types by size. An acute angle measures more than 0 and less than 90 degrees. A right angle is exactly 90 degrees and is marked with a small square at the vertex. An obtuse angle is greater than 90 and less than 180 degrees. A straight angle is exactly 180 degrees, and its two sides form a straight line. Memorizing the four boundaries, 0, 90, 180, and 360, keeps the categories clear.

The Angle Addition Postulate

The Angle Addition Postulate is the angle version of the Segment Addition Postulate: if ray BD lies in the interior of angle ABC, then the two smaller angles add to the whole, so m<ABD + m<DBC = m<ABC. A ray that splits an angle into two equal angles is an angle bisector, and there each smaller angle is exactly half of the original.

Special angle pairs

Two angles are complementary if their measures add to 90 degrees, and supplementary if their measures add to 180 degrees. A memory trick: the C in complementary can remind you of a Corner (90 degrees), and the S in supplementary can remind you of a Straight line (180 degrees). Adjacent angles share a vertex and a side but have no interior points in common; they sit side by side. A linear pair is two adjacent angles whose outer sides form a straight line, and a linear pair is always supplementary. Finally, when two lines cross, the two angles directly opposite each other are vertical angles, and vertical angles are always congruent (equal in measure).

Pair	Rule
Complementary	measures add to 90 degrees
Supplementary	measures add to 180 degrees
Linear pair	adjacent and supplementary (sum 180)
Vertical	opposite at a crossing; always equal

Worked example 1: a complement

Two angles are complementary, and one measures 34 degrees. Find the other. Step 1: complements add to 90, so the other is 90 - 34. Step 2: subtract to get 56 degrees. Check: 34 + 56 = 90, so the pair is indeed complementary.

Worked example 2: a supplement with a variable

An angle of (2x + 10) degrees is supplementary to an angle of x degrees. Find both angles. Step 1: supplements add to 180, so (2x + 10) + x = 180. Step 2: combine like terms to get 3x + 10 = 180. Step 3: subtract 10 from both sides to get 3x = 170. Step 4: divide by 3 to get x about 56.67 degrees, so the larger angle is about 2(56.67) + 10 = 123.33 degrees. Check: 56.67 + 123.33 = 180, so the two angles are supplementary.

Worked example 3: using vertical angles

Two lines cross. One angle at the crossing measures 118 degrees. Find the angle vertical to it and an angle adjacent to it. Step 1: the vertical angle is equal, so it is 118 degrees. Step 2: an adjacent angle forms a linear pair with the 118-degree angle, so it is 180 - 118 = 62 degrees. This shows the four angles at any crossing come in two equal pairs that sum to 180.

Common misconceptions

• Swapping complementary and supplementary. Complementary sums to 90; supplementary sums to 180. Keep the C-corner and S-straight hints in mind.
• Naming the wrong vertex. In angle ABC the vertex is always the middle letter B, not the first letter.
• Assuming adjacent means supplementary. Adjacent angles only sum to 180 when they also form a linear pair (their outer sides make a straight line).
• Thinking vertical angles are supplementary. Vertical angles are equal, not supplementary. The pairs that sum to 180 at a crossing are the linear pairs.
• Forgetting the interior condition. The Angle Addition Postulate needs the middle ray to lie inside the larger angle.

Recap

An angle is two rays from a shared vertex, measured in degrees. Acute is under 90, right is exactly 90, obtuse is between 90 and 180, and straight is 180. The Angle Addition Postulate lets parts add to a whole, and a bisector cuts an angle in half. Complementary angles sum to 90, supplementary angles sum to 180, a linear pair is adjacent and supplementary, and vertical angles are always equal.

Sources

1. OpenStax, Prealgebra 2e, Chapter 9 (Use Properties of Angles). Available free at openstax.org.
2. Khan Academy, "Geometry: Angles" unit (angle types, complementary and supplementary, vertical angles). khanacademy.org.
3. Euclid, Elements, Book I, Definitions 8 through 12 and Proposition 15 (vertical angles are equal). Clark University online edition.

Geometry is not just about shapes; it is about reasoning carefully from what we know to what must follow. Learning the language of logic here will make every proof later in the course far easier. A conditional statement has the form "if p, then q." The part after "if" is the hypothesis and the part after "then" is the conclusion. For example, "If an angle measures 90 degrees, then it is a right angle" has hypothesis "an angle measures 90 degrees" and conclusion "it is a right angle." Many everyday sentences are secretly conditionals, so a useful skill is rewriting a statement in clean if-then form before analyzing it.

Truth value and counterexamples

A conditional is true when the conclusion always follows from the hypothesis, and false when the hypothesis can be met while the conclusion fails. A single counterexample, one case where the hypothesis is true but the conclusion is false, is enough to prove a conditional false. For instance, "If a number is divisible by 3, then it is divisible by 6" is false, and 9 is a counterexample: 9 is divisible by 3 but not by 6.

Related conditionals

From one conditional we can build three related statements:

• The converse swaps the two parts: "if q, then p."
• The inverse negates both parts: "if not p, then not q."
• The contrapositive swaps and negates: "if not q, then not p."

A key fact: a conditional and its contrapositive are always logically equivalent, meaning they are true or false together. The converse and inverse are also equivalent to each other, but not necessarily to the original. When a conditional and its converse are both true, we combine them into a biconditional: "p if and only if q," often abbreviated "iff." Good definitions are always biconditional, because a definition must work in both directions.

Two kinds of reasoning

Inductive reasoning looks at examples or patterns and makes a general guess, called a conjecture. Seeing 2, 4, 6, 8 and predicting 10 is inductive. Inductive reasoning suggests ideas but does not prove them, because a pattern can break. Deductive reasoning instead starts from accepted facts, definitions, and rules and reaches a conclusion that must be true. Proofs in geometry use deductive reasoning.

Two laws of deduction

Two rules let us chain conditionals together. The Law of Detachment: if "if p, then q" is true and p is true, then q must be true. The Law of Syllogism: if "if p, then q" and "if q, then r" are both true, then "if p, then r" is true, much like linking a chain end to end.

Worked example 1: the contrapositive

Given the true statement "If it is a square, then it has four right angles," write the contrapositive. Step 1: identify p = "it is a square" and q = "it has four right angles." Step 2: the contrapositive swaps and negates, giving "if not q, then not p." Step 3: write it in words: "If it does not have four right angles, then it is not a square." Because the contrapositive is logically equivalent to the original, it is also true.

Worked example 2: the Law of Syllogism

Suppose these two statements are true: "If it is raining, then the game is canceled," and "If the game is canceled, then we study instead." Chain them with the Law of Syllogism. The hypothesis of the first is p = "it is raining," and the conclusion of the second is r = "we study instead." So we may conclude: "If it is raining, then we study instead." The shared middle statement, "the game is canceled," links the two.

Common misconceptions

• Believing the converse is automatically true. A true conditional does not make its converse true. "If a shape is a square, then it is a rectangle" is true, but the converse is false.
• Thinking many examples prove a statement. Examples support a conjecture but never prove it; only deductive reasoning proves a general claim.
• Forgetting one counterexample is enough. To disprove a conditional you need just a single case where the hypothesis holds and the conclusion fails.
• Mixing up inverse and contrapositive. The inverse negates only; the contrapositive negates and swaps. Only the contrapositive is guaranteed equivalent to the original.

Recap

A conditional is "if p (hypothesis), then q (conclusion)." Its converse swaps, its inverse negates, and its contrapositive does both; the contrapositive always matches the original truth value. A counterexample disproves a conditional. Inductive reasoning generalizes from patterns; deductive reasoning proves from accepted facts. The Law of Detachment and the Law of Syllogism let us draw certain conclusions from true conditionals.

Sources

1. Khan Academy, "Geometry foundations: Deductive and inductive reasoning" and "Conditional statements." khanacademy.org.
2. OpenStax, Elementary Algebra 2e, Appendix on logical reasoning and problem-solving strategies. Available free at openstax.org.
3. Euclid, Elements, Book I, Common Notions 1 through 5 (the foundational accepted truths used in deduction). Clark University online edition.

A proof is a logical argument that shows a statement is true beyond any doubt. This is the heart of geometry, and it may be new, so we will take it slowly. The most common school format is the two-column proof: the left column lists statements, and the right column gives the reason that justifies each statement. Every reason must be a given fact, a definition, a postulate, or a previously proven theorem. You always start from the given information and end at exactly what you were asked to prove. Nothing may be assumed just because a picture looks a certain way.

Properties of equality you may cite as reasons

Because geometry uses equations, you may cite the algebraic properties of equality as reasons. The Reflexive Property says any quantity equals itself (a = a), which is how a shared side or angle enters a proof. The Symmetric Property says if a = b then b = a. The Transitive Property says if a = b and b = c, then a = c. The Substitution Property lets you replace a quantity with an equal one. The Addition, Subtraction, Multiplication, and Division Properties of Equality let you do the same operation to both sides of an equation.

A plan before you write

Good proofs start with a plan. First, mark the given information on the diagram. Second, recall definitions that turn words into equations, such as "midpoint means two equal halves." Third, look for a shared side or angle you can claim by the Reflexive Property. Fourth, decide which theorem or postulate connects your givens to the goal, and only then fill in the two columns in order.

Worked example 1: a midpoint proof

Given: M is the midpoint of segment AB. Prove: AM = (1/2)AB.

Statements	Reasons
1. M is the midpoint of AB	1. Given
2. AM = MB	2. Definition of midpoint
3. AM + MB = AB	3. Segment Addition Postulate
4. AM + AM = AB	4. Substitution (step 2 into step 3)
5. 2 times AM = AB	5. Combine like terms
6. AM = (1/2)AB	6. Division Property of Equality

Read it top to bottom: each statement follows from the ones above it, and each has an airtight reason. That is the whole idea of a proof - no gaps, no guessing.

Worked example 2: the Vertical Angles Theorem

A classic result is the Vertical Angles Theorem: vertical angles are congruent. Given: angles 1 and 2 are vertical angles formed by two crossing lines, with angle 3 the angle adjacent to both. Prove: m<1 = m<2.

Statements	Reasons
1. Lines cross forming angles 1, 2, and 3	1. Given
2. m<1 + m<3 = 180	2. Angles 1 and 3 form a linear pair
3. m<2 + m<3 = 180	3. Angles 2 and 3 form a linear pair
4. m<1 + m<3 = m<2 + m<3	4. Substitution (both equal 180)
5. m<1 = m<2	5. Subtraction Property of Equality

Each vertical angle is supplementary to the same neighboring angle, so subtracting that shared angle forces the two vertical angles to be equal.

Common misconceptions

• Assuming from the picture. Angles that look equal or lines that look parallel prove nothing. Every claim needs a stated reason.
• Leaving a reason blank. Every statement, including the last one, needs a justification. "It just is" is never a reason.
• Skipping steps. A reader must follow each line from the ones above it. Combining two ideas into one line without a reason breaks the chain.
• Confusing a postulate with a theorem. A postulate is accepted without proof; a theorem has already been proven. Both may be cited, but know which is which.
• Reversing the given and the prove. You may use the given freely, but you may never assume the thing you are trying to prove.

Recap

A two-column proof lists statements on the left and their reasons on the right, moving from the given to the prove with no gaps. Reasons come from givens, definitions, postulates, theorems, and the properties of equality (reflexive, symmetric, transitive, substitution, and the four operation properties). Plan first by marking the diagram and recalling definitions. The midpoint proof and the Vertical Angles Theorem show the format in action.

Sources

1. Khan Academy, "Geometry: Congruence" unit, lessons on proofs and the properties of equality. khanacademy.org.
2. OpenStax, Elementary Algebra 2e, Chapter 2 (Properties of equality used to solve equations). Available free at openstax.org.
3. Euclid, Elements, Book I, Proposition 15 (vertical angles are equal) and Common Notion 3 (equals subtracted from equals are equal). Clark University online edition.

When a third line, called a transversal, crosses two other lines, it creates eight angles. When the two lines are parallel (they never meet and stay the same distance apart, written with matching arrowheads), these eight angles fall into neat, predictable relationships. Learning them lets you find every angle from just one.

The angle pairs

Using the diagram, here are the relationships when the lines are parallel:

• Corresponding angles sit in the same position at each crossing (for example 1 and 5). They are congruent.
• Alternate interior angles lie between the lines on opposite sides of the transversal (3 and 6). They are congruent.
• Alternate exterior angles lie outside the lines on opposite sides (1 and 8). They are congruent.
• Co-interior (same-side interior) angles lie between the lines on the same side (3 and 5). They are supplementary, adding to 180 degrees.

Worked example 1: reasoning out every angle

In the figure, suppose m<1 = 110 degrees. Step 1: angles 1 and 5 are corresponding, so m<5 = 110 degrees. Step 2: angle 3 is a co-interior partner of angle 5, so m<3 = 180 - 110 = 70 degrees. Step 3: angle 6 is the alternate interior partner of angle 3, so m<6 = 70 degrees as well. Step 4: angle 2 is a linear pair with angle 1, so m<2 = 180 - 110 = 70 degrees. From a single angle you can reason out all eight.

Worked example 2: solving with algebra

Alternate interior angles measure (3x + 15) and (5x - 25) degrees. Step 1: alternate interior angles are congruent, so set them equal: 3x + 15 = 5x - 25. Step 2: subtract 3x from both sides to get 15 = 2x - 25. Step 3: add 25 to both sides to get 40 = 2x. Step 4: divide by 2 to get x = 20. Step 5: substitute back: each angle is 3(20) + 15 = 75 degrees, and the check 5(20) - 25 = 75 agrees.

Worked example 3: a co-interior equation

Two co-interior (same-side interior) angles measure (2x + 30) and (4x) degrees. Step 1: co-interior angles on parallel lines are supplementary, so their sum is 180: (2x + 30) + 4x = 180. Step 2: combine like terms to get 6x + 30 = 180. Step 3: subtract 30 to get 6x = 150. Step 4: divide by 6 to get x = 25. Step 5: the angles are 2(25) + 30 = 80 and 4(25) = 100 degrees, and 80 + 100 = 180, confirming they are supplementary.

Proving lines are parallel

These relationships also work in reverse and are called the converse theorems. If a transversal makes a pair of corresponding angles congruent, or a pair of alternate interior angles congruent, or a pair of co-interior angles supplementary, then the two lines must be parallel. This is how builders check that two edges are truly parallel: measure the angles a crossing board makes and see whether the required relationship holds.

Common misconceptions

• Applying the theorems without parallel lines. Corresponding and alternate angles are only equal when the two lines are parallel. If the lines are not parallel, no such rule applies.
• Making co-interior angles equal. Same-side interior angles are supplementary (sum 180), not congruent. Only alternate and corresponding pairs are equal.
• Confusing alternate interior with alternate exterior. Interior angles lie between the two lines; exterior angles lie outside them. Both alternate pairs are congruent, but be sure which region you are in.
• Assuming lines look parallel. To conclude lines are parallel you must show an angle relationship, not rely on the drawing.

Recap

A transversal cuts two lines into eight angles. When the lines are parallel, corresponding angles, alternate interior angles, and alternate exterior angles are congruent, while co-interior (same-side interior) angles are supplementary. From one angle you can find all eight. The converse theorems let you prove two lines parallel by showing the matching angle relationship holds.

Sources

1. Khan Academy, "Geometry: Angle relationships with parallel lines and a transversal." khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (angle relationships and parallel lines). Available free at openstax.org.
3. Euclid, Elements, Book I, Propositions 27, 28, and 29 (parallel lines and the angles a transversal makes). Clark University online edition.

A triangle is a closed figure with three sides and three angles. Triangles are classified two ways. By sides: a scalene triangle has no equal sides, an isosceles triangle has at least two equal sides, and an equilateral triangle has all three sides equal. By angles: an acute triangle has all angles less than 90 degrees, a right triangle has one 90-degree angle, and an obtuse triangle has one angle greater than 90 degrees.

The Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem is one of the most useful facts in geometry: the three interior angles of any triangle add to 180 degrees. In the picture, x + y + z = 180. So if two angles are known, the third is forced. This is why a triangle can have at most one right angle or at most one obtuse angle; two such angles would already meet or exceed 180 degrees on their own.

Worked example 1: a missing angle

A triangle has angles of 50 and 70 degrees. Find the third. Step 1: the three angles sum to 180, so the third is 180 - 50 - 70. Step 2: subtract to get 180 - 120 = 60 degrees. Check: 50 + 70 + 60 = 180, so the answer is correct.

Worked example 2: angles in a ratio

The angles of a triangle are x, 2x, and 3x degrees. Find them. Step 1: their sum is 180, so x + 2x + 3x = 180. Step 2: combine like terms to get 6x = 180. Step 3: divide by 6 to get x = 30. Step 4: the three angles are 30, 2(30) = 60, and 3(30) = 90 degrees. Because one angle is 90 degrees, this is a right triangle, and 30 + 60 + 90 = 180 checks.

The isosceles triangle

An isosceles triangle has a special angle rule too. The Isosceles Triangle Theorem says the two angles opposite the equal sides, called the base angles, are themselves equal. So if an isosceles triangle has a top angle of 40 degrees, the two base angles share the remaining 180 - 40 = 140 degrees equally, giving 140 / 2 = 70 degrees each. An equilateral triangle, having all sides equal, therefore has all three angles equal to 180 / 3 = 60 degrees.

The Exterior Angle Theorem

If you extend one side of a triangle, you form an exterior angle outside the triangle. The Exterior Angle Theorem says this exterior angle equals the sum of the two remote (non-adjacent) interior angles. Worked example: if the two far interior angles are 55 and 65 degrees, the exterior angle is 55 + 65 = 120 degrees. You can check this: the exterior angle and its neighbor form a linear pair adding to 180, and the neighbor is 180 - 120 = 60 degrees, which is exactly 180 - 55 - 65, the third interior angle. Everything is consistent.

Common misconceptions

• Thinking a triangle can have two right angles. Two 90-degree angles already total 180, leaving nothing for the third, so it is impossible.
• Assuming equilateral and isosceles are separate. Every equilateral triangle is also isosceles, since it certainly has at least two equal sides.
• Adding the exterior angle to the wrong interior angles. The Exterior Angle Theorem uses the two remote (far) interior angles, not the adjacent one next to it.
• Forgetting base angles are equal. In an isosceles triangle the angles opposite the equal sides are equal, a fact easy to overlook when solving.

Recap

Triangles are classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse). The Triangle Angle Sum Theorem guarantees the interior angles total 180 degrees, so two known angles fix the third. Isosceles base angles are equal, and each angle of an equilateral triangle is 60 degrees. The Exterior Angle Theorem says an exterior angle equals the sum of the two remote interior angles.

Sources

1. Khan Academy, "Geometry: Triangle angles" (angle sum and exterior angle theorem). khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (properties of triangles). Available free at openstax.org.
3. Euclid, Elements, Book I, Proposition 32 (the interior angles of a triangle equal two right angles) and Proposition 5 (isosceles base angles are equal). Clark University online edition.

Two triangles are congruent when they have exactly the same size and shape, so one could be placed perfectly on the other. Written triangle ABC is congruent to triangle DEF, this means all three pairs of sides and all three pairs of angles match. Remarkably, you do not have to check all six pairs. Three well-chosen pairs are enough. The order of letters matters: matching letters name corresponding parts.

The three criteria in this course

• SSS (Side-Side-Side): if all three pairs of corresponding sides are equal, the triangles are congruent.
• SAS (Side-Angle-Side): if two pairs of sides and the included angle between them are equal, the triangles are congruent. The angle must sit between the two sides.
• ASA (Angle-Side-Angle): if two pairs of angles and the included side between them are equal, the triangles are congruent.

A caution: SSA (two sides and a non-included angle) does not guarantee congruence, and neither does AAA (equal angles only give similar, not congruent, triangles). Choosing the right criterion is the heart of a congruence proof.

Worked example and CPCTC

Suppose triangles ABC and ADC share side AC, with AB = AD and BC = DC. Which criterion proves them congruent? We have AB = AD (side), BC = DC (side), and AC = AC by the Reflexive Property (side). That is three pairs of sides, so the triangles are congruent by SSS.

Once triangles are proven congruent, CPCTC - Corresponding Parts of Congruent Triangles are Congruent - lets you declare any remaining pair of parts equal. In the example above, after proving the triangles congruent we could conclude m<BAC = m<DAC, which shows ray AC bisects angle BAD. CPCTC is often the final step that delivers what a proof set out to show.

Worked example: a full two-column congruence proof

Given: AB = AD and ray AC bisects angle BAD. Prove: triangle ABC is congruent to triangle ADC.

Statements	Reasons
1. AB = AD	1. Given
2. Ray AC bisects angle BAD	2. Given
3. m<BAC = m<DAC	3. Definition of angle bisector
4. AC = AC	4. Reflexive Property
5. Triangle ABC is congruent to triangle ADC	5. SAS (steps 1, 3, 4)

Notice the order in step 5: side AB, included angle at A, side AC. The equal angle sits between the two equal sides, which is exactly what SAS requires.

How to choose the right criterion

Read what you are given and count sides (S) and angles (A) that you can justify, then check their arrangement. Three sides means SSS. Two sides with the angle between them means SAS. Two angles with the side between them means ASA. If the angle or side is not in the between position, the shortcut may not apply. Always look for a shared side or a pair of vertical angles, because those give you a free congruent part by the Reflexive Property or the Vertical Angles Theorem.

Common misconceptions

• Using SSA or AAA as proofs. Two sides and a non-included angle (SSA) and three angles (AAA) do not guarantee congruence. AAA only gives similarity.
• Ignoring the included position. SAS needs the angle between the two sides, and ASA needs the side between the two angles. The wrong position can fail.
• Mislabeling corresponding parts. The order of letters in the congruence statement must match, so that corresponding vertices line up.
• Using CPCTC too early. CPCTC only applies after the triangles are already proven congruent; it cannot be the reason that proves them congruent.

Recap

Congruent triangles have identical size and shape, and three well-chosen pairs prove it. SSS uses three sides, SAS uses two sides and the included angle, and ASA uses two angles and the included side. SSA and AAA do not prove congruence. A shared side (Reflexive Property) or vertical angles often supply the third pair. Once triangles are congruent, CPCTC lets you conclude any remaining parts are equal.

Sources

1. Khan Academy, "Geometry: Congruence" unit (SSS, SAS, ASA, and CPCTC). khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (congruent triangles). Available free at openstax.org.
3. Euclid, Elements, Book I, Propositions 4 (SAS), 8 (SSS), and 26 (ASA). Clark University online edition.

Two figures are similar when they have the same shape but not necessarily the same size, written triangle ABC ~ triangle DEF. In similar figures, corresponding angles are equal and corresponding sides are proportional, meaning their lengths share a constant ratio called the scale factor. If one triangle is twice as big as another, the scale factor is 2 and every side is twice as long, while all angles stay the same.

Proving triangles similar

The easiest test is AA (Angle-Angle) similarity: if two angles of one triangle equal two angles of another, the triangles are similar. This works because the third angles are then forced equal by the angle sum. There is also SSS similarity (all three side ratios equal) and SAS similarity (two proportional sides with an equal included angle).

Solving with proportions

Because corresponding sides are proportional, we set up a proportion and solve by cross-multiplying. Worked example: triangle ABC ~ triangle DEF, with AB = 6, DE = 9, and BC = 8. Find EF. Matching sides give AB/DE = BC/EF, so 6/9 = 8/EF. Cross-multiply: 6 times EF = 9 times 8 = 72, so EF = 12.

Worked example: two-step setup with the scale factor

Triangle ABC ~ triangle DEF with AB = 6, DE = 9, and BC = 8. Find EF using the scale factor. Step 1: the scale factor from the small to the large triangle is DE/AB = 9/6 = 1.5. Step 2: multiply the matching small side by the scale factor: EF = BC times 1.5 = 8 times 1.5 = 12. This matches the cross-multiplication method above, and it shows the scale factor is just the constant that stretches every side.

Indirect measurement in the real world

A powerful real use of similarity is indirect measurement, finding a length you cannot reach. A 6-foot person casts a 4-foot shadow at the same moment a flagpole casts a 20-foot shadow. The sun's rays strike both at the same angle, so the two triangles are similar and the ratio height/shadow is the same for both. Step 1: write the proportion 6/4 = h/20. Step 2: cross-multiply to get 4h = 6 times 20 = 120. Step 3: divide by 4 to get h = 30 feet. Surveyors and foresters use exactly this idea to measure trees, buildings, and canyons.

Perimeter and area of similar figures

When two figures are similar with scale factor k, their perimeters also have ratio k, but their areas have ratio k squared. For example, if a photo is enlarged by a scale factor of 3, its border grows 3 times longer, yet it uses 3 squared = 9 times as much paper. Remembering that area scales by the square of the scale factor prevents a very common error.

Common misconceptions

• Thinking similar means same size. Similar figures share the same shape but usually differ in size; congruent is the special case where the scale factor is 1.
• Matching sides in the wrong order. Set up proportions using corresponding sides, guided by the order of letters in the similarity statement.
• Scaling area by the plain scale factor. Area grows by the square of the scale factor, not the scale factor itself.
• Believing AAA is not enough for similarity. For similarity, equal angles are plenty; in fact just two equal angle pairs (AA) already prove triangles similar.

Recap

Similar figures have equal corresponding angles and proportional corresponding sides, with the constant ratio called the scale factor. AA similarity (two equal angle pairs) is the quickest test, and proportions solved by cross-multiplication find unknown sides. Similar triangles power indirect measurement of tall objects. Perimeter scales by the scale factor k, while area scales by k squared.

Sources

1. Khan Academy, "Geometry: Similarity" unit (similar triangles, AA criterion, and solving with proportions). khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (similar figures and proportions). Available free at openstax.org.
3. Euclid, Elements, Book VI, Propositions 2 and 4 (proportional sides and equiangular, that is similar, triangles). Clark University online edition.

In a right triangle, the two sides that form the right angle are the legs, and the longest side, opposite the right angle, is the hypotenuse. The Pythagorean theorem connects them: if the legs are a and b and the hypotenuse is c, then a squared + b squared = c squared. It is one of the oldest and most used results in all of mathematics.

Finding a missing side

Worked example, finding the hypotenuse: a right triangle has legs 3 and 4. Then c squared = 3 squared + 4 squared = 9 + 16 = 25, so c = 5. Worked example, finding a leg: the hypotenuse is 13 and one leg is 5. Then 5 squared + b squared = 13 squared, so 25 + b squared = 169, giving b squared = 144 and b = 12. Always place the hypotenuse alone on the c side of the equation.

Some right triangles use whole numbers so often they are worth memorizing. These Pythagorean triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25, along with their multiples such as 6-8-10 (which is 3-4-5 doubled).

Legs	Hypotenuse	Check
3, 4	5	9 + 16 = 25
5, 12	13	25 + 144 = 169
8, 15	17	64 + 225 = 289

The converse

The converse of the Pythagorean theorem works backward: if a squared + b squared = c squared for the three sides of a triangle, then the triangle is a right triangle. For example, sides 9, 12, 15 satisfy 81 + 144 = 225 = 15 squared, so that triangle has a right angle. If the two smaller squares add to less than the largest square, the triangle is obtuse; if they add to more, it is acute.

Worked example: classifying a triangle with the converse

Is the triangle with sides 6, 8, 11 right, acute, or obtuse? Step 1: identify the longest side, 11, and compare squares. Step 2: add the two smaller squares: 6 squared + 8 squared = 36 + 64 = 100. Step 3: square the longest side: 11 squared = 121. Step 4: compare. Since 100 < 121, the two smaller squares add to less than the largest square, so the triangle is obtuse. (Had the sums been equal it would be right; had 100 exceeded 121 it would be acute.)

The distance between two points

The Pythagorean theorem also measures distance on a coordinate grid. To find the distance between two points, treat the horizontal gap as one leg and the vertical gap as the other, then the straight-line distance is the hypotenuse. Worked example: the distance from (1, 2) to (4, 6). Step 1: the horizontal leg is 4 - 1 = 3 and the vertical leg is 6 - 2 = 4. Step 2: apply the theorem: distance squared = 3 squared + 4 squared = 9 + 16 = 25. Step 3: take the square root: distance = 5. This is the distance formula in disguise.

A real-world application

A ladder leans against a wall. Its base is 5 feet from the wall and it reaches 12 feet up. How long is the ladder? The wall and ground form the right angle, so the ladder is the hypotenuse: c squared = 5 squared + 12 squared = 25 + 144 = 169, giving c = 13 feet. Builders, painters, and navigators lean on this theorem constantly.

Common misconceptions

• Putting a leg where the hypotenuse belongs. The hypotenuse is always the longest side, alone on the c side of the equation. Mixing it up gives a wrong answer.
• Forgetting to take the square root. Solving gives c squared; you must take the square root to get c itself.
• Applying the theorem to non-right triangles. The theorem holds only for right triangles. The converse is what tests whether a triangle is right.
• Adding sides instead of squares. The rule uses squares of the sides, so 3 + 4 is not the hypotenuse; 3 squared + 4 squared under a root is.

Recap

In a right triangle the legs a and b and hypotenuse c satisfy a squared + b squared = c squared. Solve for any missing side, keeping the hypotenuse alone on the c side and remembering the square root. Common triples like 3-4-5 and 5-12-13 speed up the work. The converse tests whether a triangle is right, acute, or obtuse, and the same idea gives the distance between two points on a grid.

Sources

1. Khan Academy, "Geometry: Right triangles and the Pythagorean theorem." khanacademy.org.
2. OpenStax, Elementary Algebra 2e, Chapter 9 (using the Pythagorean theorem and square roots). Available free at openstax.org.
3. Euclid, Elements, Book I, Proposition 47 (the Pythagorean theorem) and Proposition 48 (its converse). Clark University online edition.

In a right triangle, the ratios of the sides depend only on the acute angles, not on the triangle's size. These fixed ratios are the trigonometric ratios: sine, cosine, and tangent. For a chosen acute angle, we label the sides relative to it: the opposite side faces the angle, the adjacent side touches it (and is not the hypotenuse), and the hypotenuse is opposite the right angle.

SOH-CAH-TOA

The memory aid SOH-CAH-TOA captures all three definitions:

• SOH: sine of the angle = Opposite / Hypotenuse.
• CAH: cosine of the angle = Adjacent / Hypotenuse.
• TOA: tangent of the angle = Opposite / Adjacent.

Worked example, finding a side: a right triangle has an acute angle of 30 degrees and a hypotenuse of 10. Find the side opposite the 30-degree angle. Use sine: sin(30 degrees) = opposite / 10. Since sin(30 degrees) = 0.5, we get opposite = 10 times 0.5 = 5. Worked example with tangent: from a point 40 feet from a tree, the angle up to the top is 35 degrees. The height is 40 times tan(35 degrees), and since tan(35 degrees) is about 0.700, the height is about 28 feet.

Finding an angle

To find an unknown angle from two known sides, use an inverse trig ratio, written sin^-1, cos^-1, or tan^-1. Worked example: a right triangle has opposite side 3 and hypotenuse 5. Then sin(angle) = 3/5 = 0.6, so angle = sin^-1(0.6), which is about 36.87 degrees. These same ideas power surveying, navigation, and construction.

Choosing the right ratio

The secret to trig problems is picking the ratio that involves the two sides you care about. First label the sides relative to your chosen angle: opposite, adjacent, hypotenuse. Then look at which two the problem gives or asks for. If they are opposite and hypotenuse, use sine. Adjacent and hypotenuse, use cosine. Opposite and adjacent, use tangent. This simple check turns most problems into one short equation.

Worked example: the 3-4-5 triangle

A right triangle has legs 3 (opposite the smaller acute angle) and 4 (adjacent), with hypotenuse 5. Find the sine, cosine, and tangent of that smaller angle, then the angle itself. Step 1: sine = opposite / hypotenuse = 3/5 = 0.6. Step 2: cosine = adjacent / hypotenuse = 4/5 = 0.8. Step 3: tangent = opposite / adjacent = 3/4 = 0.75. Step 4: the angle is tan^-1(0.75), which is about 36.87 degrees. Notice sine and cosine each land between 0 and 1, a useful sanity check, while a tangent can be any positive value.

Angles of elevation and depression

Two special terms appear in real problems. An angle of elevation is measured upward from the horizontal to a higher object, and an angle of depression is measured downward from the horizontal to a lower object. Worked example: standing 40 feet from a tree, the angle of elevation to the top is 35 degrees. The height is 40 times tan(35 degrees), and since tan(35 degrees) is about 0.700, the tree is about 28 feet tall. Because the ground is horizontal, the angle of elevation from you equals the angle of depression from the treetop back down to you.

Common misconceptions

• Mislabeling opposite and adjacent. These labels depend on the chosen angle. Re-check them each time you switch angles.
• Calculator in the wrong mode. For degree answers the calculator must be in degree mode; radian mode gives very different numbers.
• Expecting sine or cosine above 1. Since the hypotenuse is the longest side, sine and cosine of an acute angle are always between 0 and 1.
• Using a plain ratio to find an angle. To get the angle from a ratio you must apply the inverse function (sin^-1, cos^-1, or tan^-1), not the ratio itself.

Recap

In a right triangle the ratios sine, cosine, and tangent depend only on the acute angle. SOH-CAH-TOA gives sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent. Pick the ratio that links the two sides in question to find a missing side, and use an inverse ratio to find a missing angle. Angles of elevation and depression apply these tools to real heights and distances.

Sources

1. Khan Academy, "Geometry: Right triangles and trigonometry" (sine, cosine, tangent, and inverse ratios). khanacademy.org.
2. Paul's Online Math Notes, "Review: Trig Functions" (right-triangle definitions of the trigonometric ratios). tutorial.math.lamar.edu.
3. OpenStax, Algebra and Trigonometry 2e, Chapter 7 (right-triangle trigonometry). Available free at openstax.org.

A polygon is a closed figure made of straight segments. A quadrilateral is a four-sided polygon, and several special types appear again and again. A parallelogram has both pairs of opposite sides parallel; its opposite sides and opposite angles are equal, and its diagonals bisect each other. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square is both a rectangle and a rhombus - four equal sides and four right angles. A trapezoid has exactly one pair of parallel sides.

Shape	Defining property
Parallelogram	both pairs of opposite sides parallel
Rectangle	parallelogram with four right angles
Rhombus	parallelogram with four equal sides
Square	four equal sides and four right angles
Trapezoid	exactly one pair of parallel sides

Interior angle sum

Any polygon can be split into triangles from one vertex, and each triangle contributes 180 degrees. A polygon with n sides splits into n - 2 triangles, so its interior angle sum is (n - 2) times 180 degrees. For a quadrilateral (n = 4) that is (4 - 2) times 180 = 360 degrees. For a pentagon (n = 5) it is (5 - 2) times 180 = 540 degrees, and for a hexagon it is 720 degrees.

Regular polygons

A regular polygon has all sides and all angles equal. Each interior angle then measures the total divided by n. Worked example: each interior angle of a regular hexagon is 720 / 6 = 120 degrees. A separate handy fact: the exterior angles of any convex polygon always add to 360 degrees, so each exterior angle of a regular polygon is 360 / n. For a regular pentagon each exterior angle is 360 / 5 = 72 degrees, and each interior angle is 180 - 72 = 108 degrees.

Worked example: working backward from an angle

A regular polygon has interior angles of 150 degrees each. How many sides does it have? Step 1: each exterior angle is the supplement, 180 - 150 = 30 degrees. Step 2: the exterior angles sum to 360, so the number of sides is 360 / 30 = 12. The polygon is a regular dodecagon. Working through the exterior angle is often faster than using the interior-sum formula directly.

The quadrilateral family tree

The special quadrilaterals form a family. A square is the most specific: it is at once a parallelogram, a rectangle, and a rhombus. A rectangle and a rhombus are each parallelograms, but a trapezoid is not a parallelogram because it has only one pair of parallel sides. Reading the family tree downward, each shape inherits the properties of the ones above it. So a square has every property of a parallelogram (opposite sides parallel and equal, diagonals bisect each other), plus the right angles of a rectangle and the equal sides of a rhombus.

Common misconceptions

• Thinking a square is not a rectangle. A square meets every requirement of a rectangle (a parallelogram with four right angles), so every square is a rectangle.
• Calling a trapezoid a parallelogram. A trapezoid has exactly one pair of parallel sides, so it is not a parallelogram.
• Multiplying instead of using (n - 2). The interior sum is (n - 2) times 180, not n times 180. Subtract 2 first.
• Forgetting exterior angles are constant. The exterior angles of any convex polygon total 360 degrees no matter how many sides it has.

Recap

A polygon is a closed figure of straight segments, and quadrilaterals include parallelograms, rectangles, rhombuses, squares, and trapezoids, each defined by its own property. The interior angles of an n-sided polygon sum to (n - 2) times 180 degrees, and a regular polygon's interior angle is that total divided by n. The exterior angles of any convex polygon always add to 360 degrees, giving 360 / n per angle in a regular polygon.

Sources

1. Khan Academy, "Geometry: Quadrilaterals" and "Angles of a polygon." khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (properties of quadrilaterals and polygon angle sums). Available free at openstax.org.
3. Euclid, Elements, Book I, Definition 22 (kinds of quadrilaterals) and Proposition 32 corollary (interior angles of polygons). Clark University online edition.

A circle is the set of all points a fixed distance from a center. That fixed distance is the radius, and a segment across the circle through the center is the diameter, which is twice the radius. A chord is any segment with both endpoints on the circle; the diameter is the longest chord. A portion of the circle itself is an arc, measured in degrees, and a full circle measures 360 degrees.

Central angles and arcs

A central angle has its vertex at the center. A central angle and the arc it cuts off (its intercepted arc) have the same measure. So a central angle of 80 degrees intercepts an arc of 80 degrees. A semicircle, cut off by a diameter, is a 180-degree arc. If several central angles fill the circle, their arcs must add to 360 degrees.

Worked example: two central angles in a circle measure 130 and 95 degrees and share no overlap. The remaining arc is 360 - 130 - 95 = 135 degrees.

Inscribed angles

An inscribed angle has its vertex on the circle and its sides are chords. The Inscribed Angle Theorem is a favorite result: an inscribed angle is half the measure of its intercepted arc. So an inscribed angle intercepting a 100-degree arc measures 100 / 2 = 50 degrees. Two important consequences follow. First, inscribed angles that intercept the same arc are equal. Second, an angle inscribed in a semicircle is always a right angle, because it intercepts a 180-degree arc and 180 / 2 = 90 degrees.

Worked example: an inscribed angle intercepts an arc of 76 degrees. The angle measures 76 / 2 = 38 degrees. Reversed: if an inscribed angle measures 25 degrees, its intercepted arc is 2 times 25 = 50 degrees.

Worked example: mixing central and inscribed angles

In a circle, a central angle and an inscribed angle both intercept the same arc, and the central angle measures 84 degrees. Find the inscribed angle. Step 1: a central angle equals its intercepted arc, so the arc is 84 degrees. Step 2: an inscribed angle is half its intercepted arc, so it is 84 / 2 = 42 degrees. In general, an inscribed angle is always half of a central angle that catches the same arc.

Chord facts

Chords add two more useful rules. First, in one circle, congruent chords cut off congruent arcs, and the reverse holds too. Second, a diameter that is perpendicular to a chord bisects that chord (cuts it into two equal pieces) and also bisects its arc. These facts let you find missing lengths inside a circle without measuring, and they combine neatly with the Pythagorean theorem when a radius and half a chord form a right triangle.

Common misconceptions

• Making the inscribed angle equal to its arc. An inscribed angle is half its intercepted arc, while a central angle equals its arc. Keep the two straight.
• Confusing radius and diameter. The diameter is twice the radius, so mixing them doubles or halves your answer by mistake.
• Forgetting the semicircle right angle. Any angle inscribed in a semicircle is 90 degrees, a fact easy to miss when a diameter is one of the chords.
• Assuming all chords are diameters. Only a chord through the center is a diameter; most chords are shorter.

Recap

A circle's radius reaches from the center to the edge, and the diameter, twice the radius, is the longest chord. A central angle equals its intercepted arc, arcs around the circle total 360 degrees, and an inscribed angle is half its intercepted arc. Consequently, inscribed angles on the same arc are equal, and an angle inscribed in a semicircle is a right angle. Congruent chords cut congruent arcs, and a perpendicular diameter bisects a chord and its arc.

Sources

1. Khan Academy, "Geometry: Circles" unit (arcs, central angles, and inscribed angles). khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (circle vocabulary and properties). Available free at openstax.org.
3. Euclid, Elements, Book III, Propositions 20 (central angle is double the inscribed angle) and 31 (angle in a semicircle is right). Clark University online edition.

Perimeter is the total distance around a flat shape, found by adding all side lengths; it uses ordinary length units. Area is the amount of surface a shape covers, measured in square units. Each common shape has a formula worth knowing.

Polygon formulas

• Rectangle: perimeter P = 2L + 2W; area A = L times W.
• Triangle: area A = (1/2) times b times h, where h is the height perpendicular to base b.
• Parallelogram: area A = b times h.
• Trapezoid: area A = (1/2)(b1 + b2) times h, averaging the two parallel sides.

Worked example: a rectangle is 12 cm by 5 cm. Perimeter is 2(12) + 2(5) = 24 + 10 = 34 cm, and area is 12 times 5 = 60 square cm. Worked example: a triangle has base 10 and height 6, so area is (1/2) times 10 times 6 = 30 square units.

The circle

For a circle we use the constant pi, about 3.14159. The distance around a circle is its circumference, C = 2 times pi times r (equivalently pi times d). The area is A = pi times r squared. Worked example: a circle has radius 7. Its circumference is 2 times pi times 7 = 14pi, about 43.98 units, and its area is pi times 7 squared = 49pi, about 153.94 square units.

Always match your units to the quantity: perimeter and circumference use length units like cm, while area uses square units like square cm. Mixing them up is one of the most common mistakes, so label every answer.

Worked example: a composite figure

Many real shapes are combinations. Find the area of a figure made of a 10 by 4 rectangle with a semicircle of diameter 4 attached to one short end. Step 1: rectangle area is 10 times 4 = 40 square units. Step 2: the semicircle has radius 4 / 2 = 2, so a full circle would have area pi times 2 squared = 4pi, and the half is 2pi, about 6.28 square units. Step 3: add the parts: total area is 40 + 2pi, about 46.28 square units. Break a composite shape into simple pieces, find each, then combine.

Worked example: finding a side from area

A triangle has area 30 square units and base 12. Find its height. Step 1: start from A = (1/2) times b times h, so 30 = (1/2) times 12 times h. Step 2: simplify the right side to 30 = 6h. Step 3: divide by 6 to get h = 5. Reversing a formula to solve for a missing measurement is a skill you will use often.

Common misconceptions

• Labeling area with plain units. Area is always in square units; forgetting the "square" is the most frequent slip.
• Using a slanted side as the height. In triangle and parallelogram area, the height is the perpendicular distance to the base, not a slanted side.
• Confusing radius and diameter in circle formulas. Circumference and area use the radius. If given the diameter, halve it first.
• Adding areas of overlapping pieces. In composite shapes, be sure the pieces do not overlap, or subtract the overlap.

Recap

Perimeter is the distance around a shape, in length units, and area is the surface it covers, in square units. Rectangles use P = 2L + 2W and A = LW; triangles use A = (1/2)bh; parallelograms use A = bh; trapezoids average the parallel sides. A circle uses C = 2 times pi times r and A = pi times r squared. Break composite figures into simple parts, and reverse a formula to find a missing side.

Sources

1. Khan Academy, "Geometry: Area and perimeter" (polygons and circles). khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (perimeter, area, and circles). Available free at openstax.org.
3. Euclid, Elements, Book I, Proposition 41 (triangle and parallelogram area) and Book XII, Proposition 2 (circle area proportional to the square of the diameter). Clark University online edition.

Solids are three-dimensional figures. Volume measures the space inside a solid, in cubic units, while surface area measures the total area of all its outer faces, in square units. A helpful pattern: for any prism or cylinder, the volume equals the area of the base times the height, V = B times h.

Volume formulas

• Rectangular prism (box): V = L times W times H.
• Cylinder: V = pi times r squared times h.
• Pyramid: V = (1/3) times B times h, where B is the base area.
• Cone: V = (1/3) times pi times r squared times h.
• Sphere: V = (4/3) times pi times r cubed.

Notice a cone is exactly one third of the cylinder with the same base and height, and a pyramid is one third of the matching prism. Worked example: a box 4 by 3 by 5 has volume 4 times 3 times 5 = 60 cubic units. Worked example: a cylinder with radius 3 and height 10 has volume pi times 3 squared times 10 = 90pi, about 282.7 cubic units. Worked example: a sphere of radius 3 has volume (4/3) times pi times 3 cubed = (4/3) times pi times 27 = 36pi, about 113.1 cubic units.

Surface area

For a rectangular prism, add the areas of all six faces: SA = 2(LW + LH + WH). Worked example: for a box 4 by 3 by 5, SA = 2(4 times 3 + 4 times 5 + 3 times 5) = 2(12 + 20 + 15) = 2(47) = 94 square units. For a cylinder, the surface area is two circular ends plus the wrapped-around side: SA = 2 times pi times r squared + 2 times pi times r times h. Worked example: a cylinder with radius 3 and height 10 has SA = 2 times pi times 9 + 2 times pi times 3 times 10 = 18pi + 60pi = 78pi, about 245.0 square units. Keep volume in cubic units and surface area in square units.

Worked example: a real-world capacity

A cylindrical water tank has radius 2 feet and height 5 feet. How much water does it hold? Step 1: use the cylinder volume, V = pi times r squared times h. Step 2: substitute: V = pi times 2 squared times 5 = pi times 4 times 5 = 20pi cubic feet. Step 3: as a decimal, 20 times 3.14159 is about 62.8 cubic feet. This is exactly how engineers size tanks, cans, and pipes.

Worked example: comparing a cone and a sphere

A cone and a sphere each have radius 3, and the cone's height is 3. Which holds more? Step 1: cone volume is (1/3) times pi times 3 squared times 3 = (1/3) times pi times 9 times 3 = 9pi cubic units. Step 2: sphere volume is (4/3) times pi times 3 cubed = (4/3) times pi times 27 = 36pi cubic units. Step 3: compare: 36pi is four times 9pi, so the sphere holds four times as much. Setting formulas side by side turns a vague question into a clear comparison.

Common misconceptions

• Mixing cubic and square units. Volume is always in cubic units and surface area in square units; label each correctly.
• Dropping the one-third for cones and pyramids. A cone or pyramid is one third of the matching cylinder or prism, so the 1/3 factor is essential.
• Cubing the diameter for a sphere. The sphere formula uses the radius cubed, not the diameter; halve the diameter first.
• Counting a face twice or missing one. A rectangular prism has six faces in three matching pairs; the 2(LW + LH + WH) pattern captures all of them exactly once.

Recap

Volume measures space inside a solid, in cubic units. A prism or cylinder is base area times height; a pyramid or cone is one third of that; a sphere is (4/3) times pi times r cubed. Surface area, in square units, adds all outer faces: 2(LW + LH + WH) for a box and two circles plus a wrapped rectangle for a cylinder. Substitute carefully, keep the correct units, and use the formulas to compare and design real objects.

Sources

1. Khan Academy, "Geometry: Solid geometry" (volume and surface area of prisms, cylinders, cones, pyramids, and spheres). khanacademy.org.
2. OpenStax, Prealgebra 2e, Chapter 9 (volume and surface area of common solids). Available free at openstax.org.
3. Euclid, Elements, Book XII, Propositions 7 (a pyramid is one third of a prism) and 10 (a cone is one third of a cylinder). Clark University online edition.