# 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 | $V=x\times x\times x=x^3$ Where $x$ is the length of a single side. | |

Cuboid | $V=\text{length}\times \text{width}\times \text{height}$ $V=lwh$ | |

Cylinder | $V=\pi\times\text{radius}^2\times \text{height}$ $V=\pi r^2h$ | |

Prism | $V=A\times l$ Where $A$ is the area of the cross-section, and $l$ is the length. | |

Sphere | $V=\dfrac{4}{3}\pi\times\text{radius}^3$ $V=\dfrac{4}{3}\pi r^3$ | |

Cone | $V=\dfrac{1}{3}\pi\times\text{radius}^2\times \text{height}$ $V=\dfrac{1}{3}\pi r^2h$ | |

Pyramid | $V=\dfrac{1}{3}\times A\times h$ Where $A$ is the base area, and $h$ is the height. | |

Frustum | $V=\text{volume of bigger cone} -\text{volume of smaller cone}$ | |

##### 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:*

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

$V=\dfrac{1}{3}\pi (5)^2(10)$

$V=\dfrac{250\pi}{3}cm^3$

*The exact volume of the cone is *$\underline{\dfrac{250\pi}{3} \space cm^3}$.