# Volume of 3D Shapes

## Volume of 3D Shapes Revision

**Volume of 3D Shapes**

The **volume** of a shape is a measure of how much space it occupies.

**Volume of Prisms**

The volume of a **prism** can be calculated as follows:

\text{Volume of prism}=\textbf{\textcolor{red}{area of cross section}} \times \textbf{\textcolor{blue}{length}}

This formula can be used for any form of **prism**, this includes, but is not limited to, cubes, cuboids and cylinders.

\text{Volume of a Cuboid}=\textbf{\textcolor{red}{Width}} \times \textbf{\textcolor{red}{Height}} \times \textbf{\textcolor{blue}{Length}}

This can be simplified for a cube since all three measurements are equal:

\text{Volume of a Cube}=\textbf{\textcolor{red}{Width}}\textcolor{red}{\boldsymbol{^3}}

\text{Volume of a Cylinder}=\textcolor{red}{\boldsymbol{\pi r^2}} \times \textbf{\textcolor{blue}{Length}}

Notice the use of \boldsymbol{\textcolor{red}{\pi r^2}} because the cross section is circular.

This can be expanded to more complex** prisms**.

**Example:** The diagram below shows a triangular prism. Work out the volume of the shape.

Firstly, find the **cross section**. As this is a triangular prism, this just entails finding the area of a triangle.

\text{Area of Cross Section}=\textcolor{red}{\frac{1}{2} \times 5 \times 6 =15} \text{ cm}^2

Thus,

\text{Volume of the prism }=\textcolor{red}{15}\times \textcolor{blue}{3.5}=52.5\text{ cm}^3 (3 sf)

**Volume of a Sphere**

The formula for the volume of a sphere, which can be found on the formula sheet, is as follows:

\text{Volume of a Sphere}=\frac{4}{3}\pi r^3

**Example: **The diagram below shows a sphere with a radius of 5 \text{ cm}

Work out the volume of the sphere.

Give your answer to 4 significant figures.

We are given that the radius is 5 \text{cm}, we simply need to substitute this into the formula.

\text{Volume of a sphere} = \frac{4}{3} \pi (5)^3 \approx 523.6 \text{ cm}^3 (3 sf)

**Volume of Cones and Pyramids**

The formula for the volume of a cone, which can be found on the formula sheet, is as follows:

\text{Volume of cone }=\frac{1}{3} \times \textcolor{red}{\pi r^2} \times \textcolor{blue}{\text{ perpendicular height}}

The formula for the volume of a pyramid, which is **not **on the formula sheet, is as follows:

**Example:** The diagram below shows a square-based pyramid.

Determine the volume of the pyramid.

In this instance, the base is square. So the area of the base is simply:

\textcolor{red}{14} \times \textcolor{red}{14} = \textcolor{red}{196} \text{ mm}^2

So,

\text{Volume of pyramid }=\frac{1}{3}\times\textcolor{red}{196}\times\textcolor{blue}{25} \approx 1633.33 \text{ mm}^3

## Volume of 3D Shapes Example Questions

**Question 1: **Shown below is a trapezoidal prism.

Calculate the prism’s volume.

**[3 marks]**

Firstly, we need to calculate the cross sectional area.

The cross section is a trapezium, so \frac{a+b}{2}h must be used.

\frac{45+60}{2} \times 20 =1050 \text{ cm}^2

Next, we must multiply by the length of the prism.

1050 \times 80 = 84000 \text{ cm}^3

**Question 2: **The diagram below shows a square-based pyramid.

Find the volume of the pyramid.

**[3 marks]**

The volume for a square-based pyramid is calculated using the following formula:

\text{Volume }=\frac{1}{3} \times \text{base} \times \text{height}

The base area is simply a square, so:

5\times5=25 \text{ cm}^2

The volume is:

\text{Volume }=\frac{1}{3} \times 25 \times 12=100\text{ m}^3

**Question 3: **Shown below is a triangle based pyramid.

The base of the pyramid has an area of 18 \text{ cm}^2. The vertical height of the pyramid is x+5 \text{ cm}^2 and the volume is 54 \text{ cm}^2

Work out the value of x

**[4 marks]**

The formula for the volume of a pyramid is:

\text{Volume }=\frac{1}{3} \times \text{base} \times \text{height}

This results in:

\frac{1}{3}\times 18 \times (x+5)=6(x+5)

Setting this equal to the known volume results in a linear equation:

6(x+5)=54

Which can be solved as follows:

6(x+5)\div6=54\div6

x+5=9

x=9-5=4\text{ cm}

**Question 4: **Shown below is a composite shape, formed by joining a hemisphere to the top of a cylinder.

The cylinder and the hemisphere share the same radius, 2.3 \text{ cm}

The height of the cylinder is 5.6 \text{ cm}

Find the volume of the compound shape.

Give your answer to **3** significant figures.

**[4 marks]**

To begin with, simply imagine this as two separate shapes. First we will find the volume of the cylinder using:

\text{Volume of a Cylinder} = \pi r^2 \times \text{ height}

This results in:

\pi \times 2.3^2 \times 5.6= 29.624 \pi \text{ cm}^3

This has been left in terms of \pi for the sake of accuracy.

Next calculate the volume of the hemisphere, which is simply half a sphere:

\text{Volume of a Hemisphere}= \frac{1}{2} \times \frac{4}{3} \pi r^3

Resulting in:

\frac{1}{2} \times \frac{4}{3} \pi \times 2.3^3=8.111\pi \text{ cm}^3

Finally, add the two volumes together:

8.111\pi + 29.624 \pi =119 \text{ cm}^3 (**3 **sf)

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