# 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 Angles in a triangle add up to 180° a + b + c = 180° Angles on a straight line Angles on a straight line add up to 180° a + b = 180° Angles around a point Angles around a point add up to 360° a + b + c + d + e = 360°

### Geometry Formulas Triangles

Angles in a triangle add up to 180°

Area of a triangle Area of a triangle

= ½ x b x h or ½ bh

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

Pythagoras' Theorem Pythagoras' theorem

a2 + b2 = c2

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

Pythagoras Theorem Worksheets

Triangle Trigonometry - sin, cos and tan Basic triangle trigonometry

sin θ = o/h

cos θ = a/h

tan θ = o/a

The Sine and Cosine Rules The Sine rule

 a  sin A = b  sin B = c  sin C

The Cosine rule

a2 = b2 + c2 - 2bc cos A

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

### Geometry Formula Sheet - Quadrilaterals

 Angles in a quadrilateral Angles in a quadrilateral add up to 360° a + b + c + d = 360° Angles in a rectangle Angles in a rectangle are all right angles (equal to 90°). Area and Perimeter of a rectangle Area of a rectangle = a x b Perimeter of a rectangle = 2a + 2b where a and b are the lengths of the two adjacent sides. Angles in a parallelogram Angles in a parallelogram opposide angles are equal (opposite sides are also equal). Also, since 2a + 2b = 360° , this means that a + b = 180° . Area of a parallelogram Area of a parallelogram = b x h = bh where b is the length of the base, and h is the perpendicular height of the 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

= 2πr or πd

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

Area of a circle The area of a circle

= πr2

where r is the radius of the circle.

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

 L = θ  360 2πr

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

Area of an arc of a circle The area A of an arc of a circle is:

 A = θ  360 πr2

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

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 - Angles in regular polygons

 Equilateral Triangle Angle: 60° Interior angles add up to 180° Square Angle: 90° Interior angles add up to 360° Pentagon Angle: 108° Interior angles add up to 540° Hexagon Angle: 120° Interior angles add up to 720° Heptagon Angle: 128.6° (to 1dp) Interior angles add up to 900° Octagon Angle: 135° Interior angles add up to 1080° Nonagon Angle: 140° Interior angles add up to 1260° Decagon Angle: 144° Interior angles add up to 1440°

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

Interior angles add up to (n-2) x 180°

Each angle must be 180(n-2)°/n

### Geometry Formula Sheet - 3D Shapes

Cubes Volume of a cube:

= a x a x a = a3

Surface area of a cube:

= 6a2

where a is the length of each side.

Cuboid Volume of a cuboid:

= length x width x height = lwh

Surface area of a cuboid:

= 2lw + 2wh + 2lh

Sphere Volume of a sphere:

 = 4/3 πr3 Surface area of a sphere: = 4πr2 where r is the radius of the sphere.

Cylinder Volume of a cylinder

= πr2h

Surface area of an open cylinder:

= 2πrh

Surface area of a closed cylinder:

= 2πrh + 2πr2 = 2πr(r+h)

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

Cone Volume of a cone

 = 1/3 πr2h

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

Surface area of a cone (including base):

= πr2 + πrs

Surface area of a cone (excluding base):

= πrs

Square Pyramid
(square-based pyramid) Volume of a square pyramid

 = 1/3 b2h

Surface area of a square pyramid (including base):

= b2 + 2bs

Surface area of a square pyramid (excluding base):

= 2bs

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.

Pyramid (general) Volume of a pyramid

= 13 (base area x height) = 13 Ah

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.

### 3D Geometric Shapes - Nets

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