Volume of 3D shapes: Formulae
In a nutshell
There are different formulae to learn for the volumes of 3D shapes.
More 3D shapes
There are two more 3D shapes that you need to be familiar with: prisms and frustums.
Prism
A prism is a 3D shape that is a 2D shape stretched in the third dimension. The 2D shape is called it's cross-section.
Examples
| | |
This is a prism with a cross-section that is a hexagon. Therefore, this shape is called a hexagonal prism. | This is a prism with a cross-section that is a triangle. Therefore, this shape is called a triangular prism. | A cylinder is also a type of prism, it has a circular cross-section. |
Frustum
A frustum is a smaller cone taken away from a bigger cone, or a cone with it's top part cut off.
Formulae for volumes of 3D shapes
Here are a list of 3D shapes and the formulae to calculate their volumes.
SHAPE | FORMULA | DIAGRAM |
Cube | V=x×x×x=x3 Where x is the length of a single side. | |
Cuboid | V=length×width×height V=lwh | |
Cylinder | V=π×radius2×height V=πr2h | |
Prism | V=A×l Where A is the area of the cross-section, and l is the length. | |
Sphere | V=34π×radius3 V=34πr3 | |
Cone | V=31π×radius2×height V=31πr2h | |
Pyramid | V=31×A×h Where A is the base area, and h is the height. | |
Frustum | V=volume of bigger cone−volume of smaller cone | |
Example 1
What is the exact volume of a cone that has a radius of 5cm and a height of 10cm?
Substitute r=5 and h=10 into the formula for the volume of a cone:
V=31πr2h
V=31π(5)2(10)
V=3250πcm3
The exact volume of the cone is 3250π cm3.