Geometry Formula Sheet

$Sally Salamander Geometry Formula sheets$

Welcome to the Math Salamanders' Geometry Formula Sheet area.

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

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser.

Geometry Formula Sheet

Here you will find our support page about different Geometry formulas, including triangles, circles, quadrilaterals and polygons, as well as 3d shapes. Using these sheets will help your child to:

know different geometric formula;
apply a range of formula to solve problems.

All the Math sheets in this section follow the Elementary Math Benchmarks.

Quicklinks to ...

Angles formulae
Triangles formulae
Quadrilaterals formulae
Circles formulae
Regular polygon formulae
3d Shapes formulae

Geometry Formula Sheet Angles

Angles in a triangle

$angles in a triangle$

Angles in a triangle add up to 180°

\[ a + b + c = 180^o \]

Angles on a straight line

$angles on a straight line$

Angles on a straight line add up to 180°

\[ a + b = 180^o \]

Angles around a point

$angles around a point$

Angles around a point add up to 360°

\[ a + b + c + d + e = 360^o \]

Convert Degrees to Radians

$convert radians to degrees image$

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?

Convert Degrees to Radians Calculator

Geometry Formulas Triangles

Angles in a triangle

$angles in a triangle$

Angles in a triangle add up to 180°

\[ a + b + c = 180^{\circ } \]

Area of a triangle

$area of a triangle$

Area of a triangle

\[ A = {1 \over 2} \times b \times h = {bh \over 2} \]

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

Pythagoras' Theorem

$Right triangle labelled$

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

Triangle Trigonometry - sine, cosine and tangent

$Trigonometry formulas triangles$

Basic triangle trigonometry

\[ sin (\theta) = {o \over h} \]

\[ cos (\theta) = {a \over h} \]

\[ tan (\theta) = {o \over a} \]

The Sine and Cosine Rules

$Geometry Formula Sheet Sine Rule Cosine Rule$

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 Formulas Triangles

Geometry Formula Sheet - Quadrilaterals

Angles in a quadrilateral

$angles in a quadrilateral$

Angles in a quadrilateral add up to 360°

\[ a + b + c + d = 360^o \]

a + b + c + d = 360°

Angles in a rectangle

$angles in a rectangle$

Angles in a rectangle are all right angles (equal to 90°).

Area and Perimeter of a rectangle

$rectangle labelled$

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?

Area and Perimeter of a Rectangle Support

Angles in a parallelogram

$angles in a parallelogram$

Angles in a parallelogram

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

\[ a + b = 180^o \]

Area of a parallelogram

$area of a parallelogram$

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?

Area of a Parallelogram

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:

Perimeter of a Circle Support

Area of a circle

$area of a circle$

The area of a circle

\[ A = \pi r^2 \]

where r is the radius of the circle.

Find out some more:

Area of a Circle

Length of an arc of a circle

$length of an arc of a circle$

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

$area of an arc$

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?

Area of a Sector Calculator

Triangle along a semicircle

$triangle inside a semicircle$

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

$regular polygon image 3$

Angle: 60°

Interior angles add up to 180°

\[ Area = { \sqrt 3 \over 4} s^2 \]

\[ Perimeter = 3s \]

Square

$regular polygon image 4$

Angle: 90°

Interior angles add up to 360°

\[ Area = s^2 \]

\[ Perimeter = 4s \]

Pentagon

$regular polygon image 5$

Angle: 108°

Interior angles add up to 540°

\[ Area = {1 \over 4} s^2 \sqrt {5(5 + 2 \sqrt 5)} \]

\[ Perimeter = 5s \]

Hexagon

$regular polygon image 6$

Angle: 120°

Interior angles add up to 720°

\[ Area = {3 \sqrt {3} \over 2} s^2 \]

\[ Perimeter = 6s \]

Heptagon

$regular polygon image 7$

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

$regular polygon image 8$

Angle: 135°

Interior angles add up to 1080°

\[ Area = 2 s^2 (1 + \sqrt 2) \]

\[ Perimeter = 8s \]

Nonagon

$regular polygon image 9$

Angle: 140°

Interior angles add up to 1260°

\[ Area = {9 \over 4} s^2 \cot {20^{\circ}} \]

\[ Perimeter = 9s \]

Decagon

$regular polygon image 10$

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:

Area of a Regular Polygon 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

$3 d shapes cube$

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?

Volume of a Cube Calculator

Cuboid

$3d geometric shapes cuboid$

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.

Looking for some more information?

Sphere

$3 dimensional shape sphere$

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.

Want some more information?

Cylinder

$geometry formula sheet cylinder$

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.

Looking for more information?

Cone

$geometry formula sheet cone$

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)

$square based pyramid volume$

Volume of a square pyramid

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

= ¹/₃ b²h

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 a Square Pyramid

Pyramid (general)

$volume of a pyramid$

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.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Geometry Cheat Sheets

Here you will find a range of geometry cheat sheets to help you answer a range of geometry questions.

The sheets contain information about angles, types and properties of 2d and 3d shapes, and also common formulas associated with 2d and 3d shapes.

Included in this page are:

images of common 2d and 3d shapes;
properties of 2d and 3d shapes;
formulas involving 2d shapes, such as area and perimeter, pythagoras' theorem, trigonometry laws, etc;
formulas involving 3d shapes about volume and surface area.

Using the sheets in this section will help you understand and answer a range of geometry questions.

$geometry cheat sheets image$

Geometry Cheat Sheets

List of Geometric Shapes

Here you will find a List of different Geometric Shapes.

Included with each shape is a small picture and a description of the properties the shape has and how it relates to other shapes.

Included in this page are the following shapes:

triangles
quadrilaterals
regular polygons
irregular polygons
a range of 3d shapes

All the printable Geometry worksheets in this section support the Elementary Math Benchmarks.

$list of geometric shapes image$

List of Geometric Shapes

3D Geometric Shapes - Nets

Here you will find our range of Free Nets for 3D Shapes.

The following printables contain nets of common 3D shapes that your child should know. Each net sheet is available both with and without tabs to aid sticking together.

Using these sheets will help your child to: