Geometry Formula Sheet

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Welcome to the Math Salamanders' Geometry Formula Sheet area.

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

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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.

Geometry Formula Sheet Angles

Angles in a triangle

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

Angles on a straight line add up to 180°

a + b = 180°

Angles around a point

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

Area of a triangle

= ½ x b x h or ½bh

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

Pythagoras' Theorem

Right triangle labelled

Pythagoras' theorem

a2 + b2 = c2

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

Triangle Trigonometry - sin, cos and tan

Trigonometry formulas triangles

Basic triangle trigonometry

sin θ = o/h

cos θ = a/h

tan θ = o/a

The Sine and Cosine Rules

Geometry Formula Sheet Sine Rule Cosine Rule

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 - 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

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

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

area of an arc

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

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 - Quadrilaterals

Angles in a quadrilateral

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

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

Area and Perimeter of a rectangle

rectangle labelled

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

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

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 - Angles in regular polygons

Equilateral Triangle

equilateral triangle angles

Angle: 60°

Interior angles add up to 180°

Square

square angles

Angle: 90°

Interior angles add up to 360°

Pentagon

2d shapes pentagon angles

Angle: 108°

Interior angles add up to 540°

Hexagon

shapes for kids hexagon

Angle: 120°

Interior angles add up to 720°

Heptagon

heptagon angles

Angle: 128.6° (to 1dp)

Interior angles add up to 900°

Octagon

2d shape octagon

Angle: 135°

Interior angles add up to 1080°

Nonagon

2 dimensional shapes nonagon

Angle: 140°

Interior angles add up to 1260°

Decagon

decagon angles

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

3 d shapes cube

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

3d geometric shapes cuboid

Volume of a cuboid:

= length x width x height = lwh

Surface area of a cuboid:

= 2lw + 2wh + 2lh

Sphere

3 dimensional shape sphere

Volume of a sphere:

= 4/3 πr3  

Surface area of a sphere:

= 4πr2

where r is the radius of the sphere.

Cylinder

geometry formula sheet 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

geometry formula sheet cone

Volume of a cone

= 1/3 πr2

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)

square based pyramid volume

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

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.

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.

List of Geometric Shapes

Here you will find a List of different Geometric Shapes. There are also properties listed about each shape.

Included in this page are the following shapes:

  • triangles
  • quadrilaterals
  • regular and irregular polygons
  • a range of 3d shapes
Along with each of the shapes, some of the most important properties have also been included.

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:

  • know the properties of different 3d shapes;
  • recognise different 2d shapes inside the 3d shapes;
  • construct a 3d shape from a net;
  • recognise 3d shapes in different orientations and sizes;
  • experience the fun and wonder of creating a 3d shape from a net.

All the free Printable Geometry Workheets in this section support the Elementary Math Benchmarks.

 

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