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Volume of 3D shapes: Formulae

Tutor: Bilal

# 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 $\underline{\dfrac{250\pi}{3} \space cm^3}$.

## FAQs - Frequently Asked Questions

### How do you find the volume of a 3D shape?

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