Geometry Formula Cheat Sheet: Complete Reference Guide

Geometry Formula Cheat Sheet

Geometry is the branch of mathematics that deals with shapes, sizes, positions, and properties of space. This comprehensive cheat sheet provides all the essential formulas you need for solving geometry problems, from basic shapes to complex 3D solids.

Why This Cheat Sheet is Essential:

Complete reference for all geometry formulas
Clear explanations with variable definitions
Practical examples for each formula
Interactive calculators for practice
Organized by shape type for easy navigation

Whether you're a student preparing for exams, a teacher creating lesson plans, or a professional needing quick references, this cheat sheet has everything you need.

Basic Geometry Concepts

Before diving into specific formulas, let's review the fundamental concepts and terminology used in geometry:

Perimeter

Total distance around a 2D shape

P = sum of all sides

Area

Amount of space inside a 2D shape

Measured in square units

Volume

Amount of space inside a 3D solid

Measured in cubic units

Surface Area

Total area of all surfaces of a 3D solid

Measured in square units

Common Geometric Terms

Vertex

Corner point where edges meet

Edge

Line segment where faces meet

Face

Flat surface of a 3D shape

Base

Bottom face of a 3D shape

Altitude

Height measured perpendicular to base

2D Shapes Formulas

Formulas for common two-dimensional shapes:

⬜

Square

s s

P = 4s
A = s²

P: Perimeter
A: Area
s: Side length

Example: Square with side 5 cm

Perimeter = 4 × 5 = 20 cm

Area = 5² = 25 cm²

▭

Rectangle

l w

P = 2(l + w)
A = l × w

P: Perimeter
A: Area
l: Length
w: Width

Example: Rectangle with l=8, w=3

Perimeter = 2(8+3) = 22 units

Area = 8 × 3 = 24 unit²

▱

Parallelogram

b h

P = 2(a + b)
A = b × h

P: Perimeter
A: Area
a, b: Side lengths
h: Height (⊥ to base)

◇

Rhombus

s d₁ d₂

P = 4s
A = (d₁ × d₂) ÷ 2

P: Perimeter
A: Area
s: Side length
d₁, d₂: Diagonal lengths

2D Shape Calculator

Select Shape

Enter dimensions (comma separated)

Select a shape and enter dimensions

Triangle Formulas

Triangles have special formulas based on their type and given information:

🔺

General Triangle

a b c h

P = a + b + c
A = ½ × b × h

P: Perimeter
A: Area
a, b, c: Side lengths
h: Height to base b

📐

Right Triangle

P = a + b + c
A = ½ × a × b
c² = a² + b²

P: Perimeter
A: Area
a, b: Legs
c: Hypotenuse

Pythagorean Theorem: In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

📏

Equilateral Triangle

P = 3s
A = (s²√3) ÷ 4
h = (s√3) ÷ 2

P: Perimeter
A: Area
s: Side length
h: Height

🧮

Heron's Formula

s = (a + b + c) ÷ 2
A = √[s(s-a)(s-b)(s-c)]

A: Area
a, b, c: Side lengths
s: Semi-perimeter

Use when: You know all three sides but not the height.

Triangle Types

Type	Sides	Angles	Special Properties
Equilateral	All sides equal	All angles 60°	Maximum symmetry
Isosceles	Two sides equal	Two angles equal	Line of symmetry
Scalene	All sides different	All angles different	No symmetry
Right	a² + b² = c²	One angle 90°	Pythagorean theorem

Circle Formulas

Circles and circular segments have formulas involving π (pi ≈ 3.14159):

⭕

Circle

• r

C = 2πr = πd
A = πr²

C: Circumference
A: Area
r: Radius
d: Diameter (d = 2r)
π: Pi ≈ 3.14159

Example: Circle with radius 7 cm

Circumference = 2π×7 ≈ 43.98 cm

Area = π×7² ≈ 153.94 cm²

◔

Sector of Circle

Arc Length = (θ ÷ 360°) × 2πr
Area = (θ ÷ 360°) × πr²

θ: Central angle in degrees
r: Radius

Example: 90° sector, r=10

Arc = (90/360)×2π×10 ≈ 15.71 units

Area = (90/360)×π×100 ≈ 78.54 unit²

◕

Annulus (Ring)

A = π(R² - r²)

A: Area
R: Outer radius
r: Inner radius

◑

Segment of Circle

A = ½r²(θ - sinθ)
(θ in radians)

A: Area
r: Radius
θ: Central angle in radians

Circle Calculator

Radius

Enter radius to calculate

3D Solids Formulas

Formulas for three-dimensional shapes including volume and surface area:

🧊

Cube

V = s³
SA = 6s²

V: Volume
SA: Surface Area
s: Side length

Example: Cube with side 4

Volume = 4³ = 64 unit³

Surface Area = 6×4² = 96 unit²

📦

Rectangular Prism

V = l × w × h
SA = 2(lw + lh + wh)

V: Volume
SA: Surface Area
l, w, h: Length, Width, Height

🔺

Pyramid

V = (1/3) × B × h
SA = B + (1/2) × P × l

V: Volume
SA: Surface Area
B: Base Area
h: Height
P: Base Perimeter
l: Slant Height

⚫

Sphere

V = (4/3)πr³
SA = 4πr²

V: Volume
SA: Surface Area
r: Radius

🥫

Cylinder

V = πr²h
SA = 2πr(h + r)

V: Volume
SA: Surface Area
r: Radius
h: Height

🎯

Cone

V = (1/3)πr²h
SA = πr(r + l)

V: Volume
SA: Surface Area
r: Radius
h: Height
l: Slant Height

🏐

Ellipsoid

V = (4/3)πabc

V: Volume
a, b, c: Semi-axes lengths

🏮

Torus (Donut)

V = 2π²Rr²
SA = 4π²Rr

V: Volume
SA: Surface Area
R: Major radius
r: Minor radius

Coordinate Geometry Formulas

Formulas involving points, lines, and shapes in the coordinate plane:

📍

Distance Formula

d = √[(x₂-x₁)² + (y₂-y₁)²]

d: Distance between points
(x₁,y₁), (x₂,y₂): Coordinates of two points

Example: Points (1,2) and (4,6)

d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5

📈

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

M: Midpoint coordinates

↗️

Slope Formula

m = (y₂-y₁)/(x₂-x₁)

m: Slope

Note: Vertical lines have undefined slope (division by zero)

📐

Line Equation Forms

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

Coordinate Geometry Calculator

Point 1 (x,y)

Point 2 (x,y)

Enter two points to calculate distance, midpoint, and slope

Angles and Lines Formulas

Formulas and relationships involving angles, lines, and polygons:

📐

Polygon Angle Sum

Sum = (n-2) × 180°
Each interior (regular) = [(n-2)×180°]/n

n: Number of sides

Example: Hexagon (n=6)

Sum = (6-2)×180° = 720°

Each interior = 720°/6 = 120°

🔢

Exterior Angles

Sum of exterior angles = 360°
Each exterior (regular) = 360°/n

📏

Arc Length (Radians)

s = rθ

s: Arc length
r: Radius
θ: Angle in radians

Conversion: 180° = π radians

To convert: radians = degrees × π/180

🧭

Trigonometric Ratios

sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent

Angle Relationships

Relationship	Description	Example
Complementary	Two angles sum to 90°	30° and 60°
Supplementary	Two angles sum to 180°	110° and 70°
Vertical	Opposite angles, equal measure	Intersecting lines
Adjacent	Share vertex and side	Next to each other

Interactive Geometry Tools

Geometry Formula Calculator

Select a shape and enter dimensions to calculate all properties automatically.

Select Shape

Select a shape to begin calculations

Challenge: A rectangular garden measures 12 meters by 8 meters. What is the area? If you want to put a fence around it, how much fencing do you need?

Solution:

Given: Length = 12 m, Width = 8 m

Area = length × width = 12 × 8 = 96 m²

Perimeter = 2(length + width) = 2(12 + 8) = 2 × 20 = 40 m

Answer: Area = 96 square meters, Fencing needed = 40 meters

Challenge: A cylindrical water tank has a radius of 3 meters and height of 10 meters. What is its volume in cubic meters? (Use π ≈ 3.14)

Solution:

Given: Radius r = 3 m, Height h = 10 m

Volume of cylinder = πr²h

V = 3.14 × (3)² × 10 = 3.14 × 9 × 10 = 282.6 m³

Answer: Volume = 282.6 cubic meters

Quick Reference Summary

Essential geometry formulas at a glance:

Shape	Perimeter/Circumference	Area	Volume	Surface Area
Square	4s	s²	-	-
Rectangle	2(l+w)	l×w	-	-
Triangle	a+b+c	½bh	-	-
Circle	2πr	πr²	-	-
Cube	-	-	s³	6s²
Rectangular Prism	-	-	l×w×h	2(lw+lh+wh)
Sphere	-	-	(4/3)πr³	4πr²
Cylinder	-	-	πr²h	2πr(h+r)
Cone	-	-	(1/3)πr²h	πr(r+l)

Memory Tips

Circle formulas: Remember "Cherry Pie is delicious" (C=πd) and "Apple Pies are too" (A=πr²)
Triangle area: "Half base times height" - always use perpendicular height
Volume formulas: Prisms: Base area × height; Pyramids & cones: ⅓ × Base area × height
Surface area: Sum of areas of all faces
Pythagorean theorem: a² + b² = c² (for right triangles only)

Table of Contents

Common Variables