Volumes of single and joined solids

How much does it hold?

A bucket, a bottle, a cricket ball and a water tank all take up space. The amount of space inside a solid is its volume. We measure it in cubic units: $\text{cm}^3$ and $\text{m}^3$. For liquids we also use the litre. One litre is $1000\ \text{cm}^3$, and one cubic metre is $1000$ litres.

The five formulas

Let $r$ be the radius and $h$ the height.

• Cuboid with edges $a$, $b$ and $c$: $V = abc$. A cube of edge $a$: $V = a^3$.
• Cylinder: $V = \pi r^2 h$. This is the area of the base times the height.
• Cone: $V = \dfrac{1}{3}\pi r^2 h$.
• Sphere: $V = \dfrac{4}{3}\pi r^3$.
• Hemisphere: $V = \dfrac{2}{3}\pi r^3$, half of the sphere.

Where does the $\frac{1}{3}$ come from? Make a hollow cone and a hollow cylinder with the same radius and the same height. Fill the cone with water and pour it into the cylinder. Do this three times and the cylinder is full. So a cone is exactly one third of the cylinder. A proof for any solid needs calculus, and you will meet it later.

Archimedes found a link for the sphere. Put a ball tightly inside a cylinder, so that the ball touches the sides, the top and the bottom. The cylinder has radius $r$ and height $2r$, so its volume is $\pi r^2 \times 2r = 2\pi r^3$. The ball takes exactly two thirds of it: $\frac{2}{3} \times 2\pi r^3 = \frac{4}{3}\pi r^3$.

A neat fact. Take a cone, a hemisphere and a cylinder with the same radius $r$ and the same height $r$. Their volumes are $\frac{1}{3}\pi r^3$, $\frac{2}{3}\pi r^3$ and $\pi r^3$. They are in the ratio $1 : 2 : 3$.

Joined solids: volumes add

Surface area loses the faces where two solids touch. Volume does not. The two solids fill separate spaces, and the space inside the combination is the space inside the first plus the space inside the second. So

• for a solid made by joining parts, add the volumes;
• for a solid with a hollow cut out, subtract the volume of the hollow.

Worked examples

Example 1. An ice cream cone. A cone of height $12\ \text{cm}$ and radius $3\ \text{cm}$ is full of ice cream. A hemisphere of ice cream with the same radius sits on top. Find the total volume of ice cream.

1. The cone. $\dfrac{1}{3}\pi \times (3\ \text{cm})^2 \times 12\ \text{cm} = 36\pi\ \text{cm}^3$.
2. The hemisphere. $\dfrac{2}{3}\pi \times (3\ \text{cm})^3 = 18\pi\ \text{cm}^3$.
3. Together. $36\pi + 18\pi = 54\pi\ \text{cm}^3$, which is about $169.6\ \text{cm}^3$. That is $0.17$ litres.

Example 2. A capsule. A capsule is a cylinder with a hemisphere on each end. It is $20\ \text{mm}$ long in all, and its radius is $3\ \text{mm}$. Find its volume.

1. The cylinder part. Its length is $20\ \text{mm} - 2 \times 3\ \text{mm} = 14\ \text{mm}$. The volume is $\pi \times (3\ \text{mm})^2 \times 14\ \text{mm} = 126\pi\ \text{mm}^3$.
2. The two ends. Two hemispheres make one sphere: $\dfrac{4}{3}\pi \times (3\ \text{mm})^3 = 36\pi\ \text{mm}^3$.
3. Together. $126\pi + 36\pi = 162\pi\ \text{mm}^3$, which is about $508.9\ \text{mm}^3$.

Example 3. A tank in litres. A cylindrical tank has radius $1.4\ \text{m}$ and depth $2\ \text{m}$. Find how many litres it holds, with $\pi = \frac{22}{7}$.

1. The volume. $\dfrac{22}{7} \times (1.4\ \text{m})^2 \times 2\ \text{m} = 12.32\ \text{m}^3$.
2. Litres. $12.32 \times 1000 = 12\,320$ litres.

Practise with new numbers

Try it in Python

This program shows a cone, a hemisphere and a cylinder with the same radius and height, and prints the ratio of their volumes. It also checks the ball in the cylinder.

Common mistakes

• Forgetting the $\frac{1}{3}$ of the cone. A cone holds one third of the cylinder with the same base and height.
• Using the slant height for the volume of a cone. Volume needs the vertical height $h$. The slant height $l$ is for the curved surface.
• Using the diameter as the radius. If a ball is $12\ \text{cm}$ across, its radius is $6\ \text{cm}$.
• Forgetting that a litre is $1000\ \text{cm}^3$. Convert cm$^3$ to litres by dividing by $1000$.

Remember these

• Cylinder $\pi r^2 h$. Cone $\frac{1}{3}\pi r^2 h$. Sphere $\frac{4}{3}\pi r^3$. Hemisphere $\frac{2}{3}\pi r^3$.
• A cone, a hemisphere and a cylinder with equal radius and height $r$ are in the ratio $1 : 2 : 3$.
• Volumes of joined parts add, and the volume of a hollow is taken away.
• $1$ litre is $1000\ \text{cm}^3$, and $1\ \text{m}^3$ is $1000$ litres.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cuboid water tank measures $2\ \text{m}$ by $1.5\ \text{m}$ by $1\ \text{m}$. How many litres does it hold?

2. A cylinder has radius $5\ \text{cm}$ and height $12\ \text{cm}$. Find its volume in terms of $\pi$.

3. A cone has radius $6\ \text{cm}$ and height $7\ \text{cm}$. Find its volume in terms of $\pi$.

4. A ball has radius $3\ \text{cm}$. Find its volume in terms of $\pi$.

5. A cone and a hemisphere both have radius $3\ \text{cm}$, and the cone is $3\ \text{cm}$ high. What is the ratio of their volumes?