High School Geometry Cheat Sheet [Printable PDF Included]

Geometry has a lot to remember: angle rules, triangle theorems, circle properties, area and volume formulas, and the reasons you need for proofs.

It’s easy to mix them up right when you need them. This post puts everything in one place, from angles and triangles to circles, coordinate geometry, transformations, and proofs.

Each section below matches what’s on the cheat sheet, so you can study from this page now and keep the PDF for quick reference later.

Scroll to the end to grab the printable PDF, designed to print front and back on a single sheet.

What’s Inside

Front side

1. Basics and Angles

2. Triangles

3. Right Triangles and Trigonometry

4. Polygons and Quadrilaterals

Back side

5. Circles

6. Coordinate Geometry

7. Area, Surface Area, and Volume

8. Transformations

9. Proof Toolkit

The Cheat Sheet, Section by Section

Front Side (Page 1): Shapes and Angles

1. Basics and Angles

Notation

Term	Notation	Meaning
Point	A	A location
Line	line AB	Extends forever in both directions
Ray	ray AB	Starts at A, passes through B, goes on forever
Segment	AB	Part of a line with two endpoints
Angle	∠ABC	Vertex at B

Angle types

Type	Measure
Acute	Less than 90°
Right	Exactly 90°
Obtuse	Between 90° and 180°
Straight	Exactly 180°
Reflex	Between 180° and 360°

Angle pairs

Pair	Rule
Complementary	Add up to 90°
Supplementary	Add up to 180°
Linear pair	Adjacent and supplementary
Vertical angles	Opposite each other and equal

Parallel lines cut by a transversal

Angle pair	Relationship
Corresponding	Equal
Alternate interior	Equal
Alternate exterior	Equal
Co-interior (same-side interior)	Supplementary (add to 180°)

Common mistake: Co-interior angles add to 180°; they are not equal. Also, these rules only work when the lines are parallel.

2. Triangles

Classification

By sides	By angles
Equilateral: 3 equal sides	Acute: all angles under 90°
Isosceles: 2 equal sides	Right: one 90° angle
Scalene: no equal sides	Obtuse: one angle over 90°

Key facts

• Angles add up to 180°.
• Exterior angle = the sum of the two remote interior angles.
• Triangle inequality: the sum of any two sides is greater than the third.
• The longest side is opposite the largest angle.

Congruence and similarity

Congruent (same size and shape)	Similar (same shape)
SSS, SAS, ASA, AAS, HL (right triangles only)	AA, SSS similarity (all three sides proportional), SAS similarity (two sides proportional, included angle equal)

If the side ratio is k, the perimeter ratio is k and the area ratio is k².

Special segments and centers

Segment	Meeting point	Notes
Median	Centroid	Splits each median 2:1 from the vertex
Altitude	Orthocenter	Perpendicular from vertex to opposite side
Angle bisector	Incenter	Center of the inscribed circle
Perpendicular bisector	Circumcenter	Center of the circumscribed circle

Area formulas

• Basic: A = ½ × base × height
• Two sides and the included angle: A = ½ab sin C
• Equilateral: A = (√3/4)s²
• Heron’s formula: A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2

Common mistake: AAA proves similarity, not congruence. SSA proves neither (except HL in right triangles).

3. Right Triangles and Trigonometry

Pythagorean theorem: a² + b² = c² (c is the hypotenuse)

Common triples (and their multiples): 3-4-5, 5-12-13, 8-15-17, 7-24-25

Special right triangles

Triangle	Side ratio
45°-45°-90°	x : x : x√2
30°-60°-90°	x : x√3 : 2x (short leg is opposite 30°)

SOH-CAH-TOA

• sin θ = Opposite / Hypotenuse
• cos θ = Adjacent / Hypotenuse
• tan θ = Opposite / Adjacent

Exact values

θ	30°	45°	60°
sin	1/2	√2/2	√3/2
cos	√3/2	√2/2	1/2
tan	√3/3	1	√3

Any triangle

• Law of sines: a/sin A = b/sin B = c/sin C
• Law of cosines: c² = a² + b² − 2ab cos C

Common mistake: “Opposite” and “adjacent” depend on which angle you’re using, but the hypotenuse never changes. Also check that your calculator is in degree mode.

4. Polygons and Quadrilaterals

Polygon formulas (n = number of sides)

Quantity	Formula
Sum of interior angles	(n − 2) × 180°
Each interior angle (regular)	(n − 2) × 180° / n
Sum of exterior angles	360°
Each exterior angle (regular)	360° / n
Number of diagonals	n(n − 3) / 2

Quadrilateral comparison

Shape	Sides	Angles	Diagonals	Area
Parallelogram	Opposite sides parallel and equal	Opposite equal; consecutive supplementary	Bisect each other	base × height
Rectangle	Opposite sides parallel and equal	All 90°	Equal; bisect each other	L × W
Rhombus	All sides equal	Opposite equal	Perpendicular; bisect each other and the angles	½ × d₁ × d₂
Square	All sides equal	All 90°	Equal; perpendicular; bisect each other	s²
Trapezoid	At least one pair of parallel sides (the bases)	Angles along each leg add to 180°	Equal only if isosceles	½(b₁ + b₂) × h
Kite	Two pairs of adjacent sides equal	One pair of opposite angles equal	Perpendicular; one bisects the other	½ × d₁ × d₂

Some courses define a trapezoid as having exactly one pair of parallel sides. Use your class’s definition.

Common mistake: A square is both a rectangle and a rhombus, but not every rectangle is a square. Rectangle diagonals are equal; rhombus diagonals are perpendicular but not equal.

Back Side (Page 2): Circles, Measurement, and Proofs

5. Circles

Parts: radius, diameter (2r), chord, secant (line through two points), tangent (touches at one point), arc, sector, segment

Formulas

Quantity	Formula
Circumference	C = 2πr = πd
Area	A = πr²
Arc length	(θ/360°) × 2πr
Sector area	(θ/360°) × πr²

Angle rules

Angle	Equals
Central angle	Its intercepted arc
Inscribed angle	½ its intercepted arc
Angle in a semicircle	90°
Angle formed by a tangent and a chord	½ its intercepted arc
Two chords crossing inside	½ (sum of the two arcs)
Two secants/tangents meeting outside	½ (far arc − near arc)

Theorems

• A tangent is perpendicular to the radius at the point of tangency.
• Two tangents from the same outside point are equal in length.
• A perpendicular from the center to a chord bisects the chord.
• Opposite angles of a cyclic quadrilateral add to 180°.

Segment lengths

• Intersecting chords: PA × PB = PC × PD (the two pieces of one chord multiplied equal the two pieces of the other)
• Two secants from one outside point: (whole secant) × (outside part) = (whole secant) × (outside part)
• Tangent and secant: tangent² = (whole secant) × (outside part)

Common mistake: A central angle equals its arc, but an inscribed angle is half its arc. Also, make sure you use the radius, not the diameter, in area formulas.

6. Coordinate Geometry

Concept	Formula
Distance	d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint	((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope	m = (y₂ − y₁)/(x₂ − x₁)
Parallel lines	Equal slopes
Perpendicular lines	Slopes multiply to −1 (negative reciprocals)
Horizontal / vertical line	Slope 0 / slope undefined

Equations of a line

Form	Equation
Slope-intercept	y = mx + b
Point-slope	y − y₁ = m(x − x₁)
Standard	Ax + By = C

Equation of a circle: (x − h)² + (y − k)² = r², with center (h, k) and radius r

Common mistake: The signs in a circle equation are flipped: (x + 3)² means h = −3. The right side is r², not r. In the slope formula, subtract the coordinates in the same order on top and bottom.

7. Area, Surface Area, and Volume

2D area

Shape	Area
Square	s²
Rectangle	L × W
Parallelogram	b × h
Triangle	½ × b × h
Trapezoid	½(b₁ + b₂) × h
Rhombus / Kite	½ × d₁ × d₂
Circle	πr²
Regular polygon	½ × apothem × perimeter

3D solids (L = length, W = width, H = height, B = base area, P = base perimeter, ℓ = slant height; all solids are right solids)

Solid	Lateral area	Total surface area	Volume
Cube	4s²	6s²	s³
Rectangular prism	2H(L + W)	2(LW + LH + WH)	L × W × H
Right prism	P × H	P × H + 2B	B × H
Cylinder	2πrH	2πr² + 2πrH	πr²H
Regular pyramid	½ × P × ℓ	B + ½ × P × ℓ	⅓ × B × H
Cone	πrℓ	πr² + πrℓ	⅓πr²H

Sphere: Surface area = 4πr², Volume = (4/3)πr³

Scale factor k: lengths × k, areas × k², volumes × k³

Common mistake: Slant height (ℓ) is not the same as height (H). For a cone, ℓ² = r² + H². Also, don’t forget the ⅓ in cone and pyramid volumes.

8. Transformations

Transformation	What it does	Rule
Translation	Slides	(x, y) → (x + a, y + b)
Reflection over x-axis	Flips	(x, y) → (x, −y)
Reflection over y-axis	Flips	(x, y) → (−x, y)
Reflection over y = x	Flips	(x, y) → (y, x)
Reflection over y = −x	Flips	(x, y) → (−y, −x)
Rotation 90° counterclockwise	Turns	(x, y) → (−y, x)
Rotation 180°	Turns	(x, y) → (−x, −y)
Rotation 270° counterclockwise	Turns	(x, y) → (y, −x)
Dilation (center at origin)	Resizes	(x, y) → (kx, ky)

Dilation: k > 1 enlarges, 0 < k < 1 shrinks.

Isometries (preserve size and shape): translations, reflections, rotations. Dilations change size, so the image is similar, not congruent.

Common mistake: A 90° clockwise rotation is the same as 270° counterclockwise. Check the direction before applying the rule.

9. Proof Toolkit

Reasons bank

Reason	When to use it
Given	Information stated in the problem
Definition of midpoint	Splits a segment into two equal parts
Definition of angle bisector	Splits an angle into two equal angles
Definition of perpendicular	Lines form 90° angles
Reflexive property	A side or angle is equal to itself (shared side)
Symmetric property	If a = b, then b = a
Transitive property	If a = b and b = c, then a = c
Substitution	Replace a quantity with an equal one
Addition / Subtraction property	Add or subtract equal amounts from equal things
Vertical angles theorem	Vertical angles are equal
Linear pair postulate	Linear pair angles are supplementary
Parallel line theorems	Corresponding, alternate interior, and co-interior angles
Triangle angle sum	Angles add to 180°
Isosceles triangle theorem	Equal sides have equal opposite angles, and vice versa
SSS, SAS, ASA, AAS, HL	Prove triangles congruent
CPCTC	Corresponding parts of congruent triangles are congruent

Two-column proof template

Statements	Reasons
1. (what you know)	1. Given
2. (next fact)	2. (definition, property, or theorem)
3. △ABC ≅ △DEF	3. (SSS, SAS, ASA, AAS, or HL)
4. (what you want to show)	4. CPCTC

Quick strategy: Mark the given information on the diagram, write down what you need to prove, then work backward to find which triangles or angles will get you there.

Common mistake: Using CPCTC before proving the triangles congruent. Also, never use the statement you’re trying to prove as one of your reasons.

Top 5 Mistakes to Avoid

These are the errors that cost students the most points. Each one is covered on the cheat sheet.

1. Mixing up central and inscribed angles. A central angle equals its arc, but an inscribed angle is half its arc. Also use the radius, not the diameter, in area formulas.
2. Using parallel-line rules when the lines aren’t parallel. Corresponding and alternate angles are equal only when the lines are parallel. Co-interior angles add to 180°; they are not equal.
3. Proving congruence with AAA or SSA. AAA proves similarity only. SSA proves nothing, except HL in right triangles.
4. Confusing slant height with height. In cone and pyramid formulas, ℓ and H are different. For a cone, ℓ² = r² + H². Don’t forget the ⅓ in cone and pyramid volumes.
5. Using CPCTC too early. You can use it only after you’ve proven the triangles congruent. Never use the statement you’re trying to prove as a reason.

How to Study With This Cheat Sheet

A cheat sheet only helps if you actually use it. Here are five ways to turn it from a handout into a study tool.

1. Annotate it

Print the sheet and write on it. Notes in your own handwriting stick better than printed text, and they make the sheet match your class.

• Add your teacher’s notation and any theorem names your class uses.
• Circle or highlight the formulas you forget most often.
• Sketch a small diagram next to each theorem. A quick drawing of a tangent meeting a radius, or of a cone with its slant height marked, is easier to recall than a sentence.
• Use one highlighter color per type of mistake (for example, yellow for formulas you forget, pink for rules you mix up).

2. Cover and quiz

Active recall beats rereading.

1. Cover the right column of a table and read only the left.
2. Say the answer out loud or write it on scrap paper.
3. Uncover the column and check yourself.
4. Put a small dot next to every item you miss.

Do one section a day, and spend extra time on the dotted items. For the formula-heavy sections (circles, 3D solids, transformations), try the reverse too: cover the left column and name the shape or rule from the formula.

3. Work one problem per section

After reviewing a section, find one homework or textbook problem that uses it. Solve it with the sheet closed, then check your work against the sheet. This shows you the difference between recognizing a formula and being able to use it.

4. Build a weekly review routine

• Day 1 to 5: Review one or two sections a day using cover and quiz.
• Day 6: Go through only the items you dotted during the week.
• Day 7: Try the Top 5 Mistakes list without looking, then check it.

Short, repeated sessions of 10 to 15 minutes work better than one long session the night before a test.

5. Keep it in your binder

Put the sheet in the front pocket or a sheet protector so it’s with you for homework, quizzes, and exam prep. If your teacher allows a reference sheet on tests, your annotated copy is the one to bring (check the rules first).

Download the Printable Geometry Cheat Sheet (PDF)

What you get:

• 2 pages (front and back of one sheet)
• A4 and Letter sizes
• Black-and-white friendly

How to Print Front and Back

1. Select double-sided / duplex printing in your print dialog.
2. Choose “Flip on long edge” so page 2 isn’t upside down on the back.
3. No duplex printer? Print page 1, then reload the same sheet and print page 2 on the back. Or print both pages and tape or glue them back to back.
4. Print at 100% / “Actual size” so the text stays readable.

Wrapping Up,

Geometry is easier when you can see every rule in one place. This cheat sheet covers angles, triangles, trigonometry, polygons, circles, coordinate geometry, area and volume, transformations, and proofs. Use it to check your homework, review before a quiz, and spot the mistakes that cost the most points.

Don’t just read it. Annotate it, quiz yourself with the right column covered, and go back to the sections you keep missing. A few minutes a day will help more than one long session the night before a test.