# Geometry Formula Sheet

Welcome to the Math Salamanders' Geometry Formula Sheet area.

Here you will find a range of different Geometry formulas for 2d and 3d shapes.

## Geometry Formula Sheet

### Geometry Formula Sheet Angles

#### Angles in a triangle add up to 180°

$a + b + c = 180^o$

#### Angles on a straight line add up to 180°

$a + b = 180^o$

#### Angles around a point add up to 360°

$a + b + c + d + e = 360^o$

#### Convert from degrees (d) to radians (r)

$r = {\pi \over 180 } d$

#### Convert from radians (r) to degrees (d)

$d = {180 \over \pi } r$

Want to find out some more?

### Geometry Formulas Triangles

#### Angles in a triangle add up to 180°

$a + b + c = 180^{\circ }$

#### Area of a triangle

$A = {1 \over 2} \times b \times h = { 1\over 2} bh \; or \; {bh \over 2}$

where b is the length of the base of the triangle and h is the perpendicular height.

#### Pythagoras' theorem

$a^2 + b^2 = c^2$

where a,b and c are the sides of a right triangle. Side c is the hypotenuse (longest side).

Pythagoras Theorem Worksheets

#### Basic triangle trigonometry

$sin (\theta) = {o \over h}$

$cos (\theta) = {a \over h}$

$tan (\theta) = {o \over a}$

#### The Sine rule

${a \over sin(A)} = {b \over sin(B)} = {c \over sin(C)}$

#### The Cosine rule

$a^2 = b^2 + c^2 - 2bc \cdot cos(A)$

For more Geometry formulas about triangles, including examples showing the sine and cosine rules, use the link below.

### Geometry Formula Sheet - Quadrilaterals

$a + b + c + d = 360^o$

a + b + c + d = 360°

#### Area of a rectangle

$A = a \times b \; or \; a \cdot b$

#### Perimeter of a rectangle

$P = 2a + 2b$

where a and b are the lengths of the two adjacent sides.

Want to know some more?

#### Angles in a parallelogram

opposide angles are equal (opposite sides are also equal).

$a + b = 180^o$

#### Area of a parallelogram

$A = b \times h$

where b is the length of the base, and h is the perpendicular height of the parallelogram.

Want to find out some more?

### Geometry Formula Sheet - Circles

#### Circumference of a circle

The circumference of a circle is the distance all the way around the outside of the circle, or the perimeter of the circle.

#### The Circumference of a circle

$C = 2 \pi r \; or \; \pi d$

where r is the radius of the circle and d is the diameter of the circle.

Find out some more:

#### The area of a circle

$A = \pi r^2$

where r is the radius of the circle.

Find out some more:

#### The length L of an arc of a circle is:

$L = { \theta \over 180} \cdot \pi r$

where θ is the angle (in degrees) and r is the radius.

Area of a sector of a circle

The area A of a sector of a circle is:

$A = { \theta \over 360} \cdot \pi r^2$

where θ is the angle (in degrees) and r is the radius.

Want to find out some more?

#### Triangle along a semicircle

A triangle drawn inside a circle with one side going along the diameter, and the other 2 sides meeting at any point along the edge of the circle will always make a right angle.

The triangle will always be a right triangle.

### Geometry Formula Sheet - Regular Polygons

 Equilateral Triangle Angle: 60° Interior angles add up to 180° $Area = { \sqrt 3 \over 4} s^2$ $Perimeter = 3s$ Square Angle: 90° Interior angles add up to 360° $Area = s^2$ $Perimeter = 4s$ Pentagon Angle: 108° Interior angles add up to 540° $Area = {1 \over 4} s^2 \sqrt {5(5 + 2 \sqrt 5)}$ $Perimeter = 5s$ Hexagon Angle: 120° Interior angles add up to 720° $Area = {3 \sqrt {3} \over 2} s^2$ $Perimeter = 6s$ Heptagon Angle: 128.6° (to 1dp) Interior angles add up to 900° $Area = {7 \over 4} s^2 \cot ({180^{\circ} \over 7})$ $Perimeter = 7s$ Octagon Angle: 135° Interior angles add up to 1080° $Area = 2 s^2 (1 + \sqrt 2)$ $Perimeter = 8s$ Nonagon Angle: 140° Interior angles add up to 1260° $Area = {9 \over 4} s^2 \cot {20^{\circ}}$ $Perimeter = 9s$ Decagon Angle: 144° Interior angles add up to 1440° $Area = {5 \over 2} s^2 \sqrt {5 + 2 \sqrt 5}$ $Perimeter = 10s$

#### Angles in a Regular Polygon

A general formula for this rule for an n-sided regular polygon is:

Interior angles add up to $(n - 2) \times 180^{\circ}$

Each angle must be ${180 (n-2)^{\circ} \over n}$

#### Area of a Regular Polygon

A general formula for the area of a regular polygon with n-side of length s is:

$A = {ns^2 \over 4 \tan ({180^{\circ} \over n})}$

This can also be written: $A = {n \over 4 } s^2 \cot ({180 \over n})^{\circ}$

For more support finding the area of regaular polygons, try our calculator:

#### Perimeter of a Regular Polygon

A general formula for the perimeter of a regular polygon with n-side of length s is:

$P = n \times s$

### Geometry Formula Sheet - 3D Shapes

Cubes

#### Volume of a cube:

$V = a \times a \times a = a^3$

#### Surface area of a cube:

$A = 6a^2$

where a is the length of each side.

Want to find out some more?

Cuboid

#### Volume of a cuboid:

$V = l \times w \times h = lwh$

#### Surface area of a cuboid:

$A = 2lw + 2wh + 2lh$

where l is the length, w is the width and h is the height of the cuboid.

Sphere

#### Volume of a sphere:

$V = {4 \over 3} \pi r^3$

#### Surface area of a sphere:

$A = 4 \pi r^2$

where r is the radius of the sphere.

Cylinder

#### Volume of a cylinder

$V = \pi r^2 h$

#### Surface area of a closed cylinder:

$A = 2 \pi r h + 2 \pi r^2 = 2 \pi r (r + h)$

#### Surface area of an open cylinder (a hollow tube):

$A = 2 \pi r h$

where r is the radius of the cylinder and h is the height.

Cone

#### Volume of a cone

$V = {1 \over 3} \pi r^2 h$

#### Surface area of a cone (including base):

$A = \pi r^2 + \pi r s = \pi r (r + s)$ OR $A = \pi r (r + \sqrt {r^2 + h^2})$

#### Surface area of a cone (excluding base):

$A = \pi r s$ OR $A = \pi r \sqrt {r^2 + h^2}$

where r is the radius of the widest part of the cone, h is the height and s is the slant height of the cone.

Want some more help?

Square Pyramid
(square-based pyramid)

#### Volume of a square pyramid

$V = {1 \over 3} b^2 h$

#### Surface area of a square pyramid (including base):

$A = b^2 + 2 b s$

#### Surface area of a square pyramid (excluding base):

$A = 2 b s$

where b is the length of one side of the base, h is the vertical height of the pyramid and s is the slant height of one of the triangles.

Looking for more support?

#### Volume of any pyramid

${1 \over 3} A h$

where A is the area of the base, and h is the height.

This formula works for any pyramid with a rectangular or triangular base and triangular sides.

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