Buckets, glasses and lampshades

A plan for every frustum problem

Most word problems about a bucket, a glass or a lampshade follow the same plan.

1. Draw the side view and write $R$, $r$ and $h$ on it. Check which radius is which.
2. Find the slant height $l = \sqrt{h^2 + (R - r)^2}$ if the question needs an area.
3. Decide which surfaces count. A bucket and a glass have the curved surface and the base. A lampshade has only the curved surface.
4. Use the right formula. Area for metal, cloth or paint. Volume for water, milk or sand.
5. Change the units. $1000\ \text{cm}^3$ is one litre.

Example 1. A bucket full of water has top radius $18\ \text{cm}$, bottom radius $6\ \text{cm}$ and height $15\ \text{cm}$. How many litres of water does it hold? Give the answer to two decimal places.

1. The volume. $\dfrac{\pi \times 15\ \text{cm}}{3}(324 + 108 + 36)\ \text{cm}^2 = 5\pi \times 468\ \text{cm}^3 = 2340\pi\ \text{cm}^3$.
2. The decimal. $2340\pi = 7351.3\ \text{cm}^3$.
3. In litres. $7351.3 \div 1000 = 7.35$ litres.

A cone cut parallel to its base

Cut a cone with a plane parallel to its base. The small cone on top is a smaller copy of the whole cone. Suppose its height is $f$ times the height of the whole cone, where $f$ is a fraction between $0$ and $1$. Every length of the small cone is $f$ times the matching length of the whole cone. So

• its curved surface is $f^2$ times the curved surface of the whole cone, because areas go with the square;
• its volume is $f^3$ times the volume of the whole cone, because volumes go with the cube.

The frustum has the rest. For $f = \frac{1}{2}$, the small cone has $\frac{1}{4}$ of the curved area and $\frac{1}{8}$ of the volume. The frustum has $\frac{3}{4}$ of the curved area and $\frac{7}{8}$ of the volume. So the frustum has 3 times the curved surface of the small cone and 7 times its volume.

Example 2. A cone is cut by a plane parallel to its base, so that the small cone at the top has height $\frac{1}{3}$ of the height of the whole cone. Find the ratio of the volume of the frustum to the volume of the small cone.

1. The small cone. Its volume is $\left(\dfrac{1}{3}\right)^3 = \dfrac{1}{27}$ of the whole cone.
2. The frustum. It has $1 - \dfrac{1}{27} = \dfrac{26}{27}$ of the whole cone.
3. The ratio. $\dfrac{26/27}{1/27} = 26$. The frustum has $26$ times the volume of the small cone.

Example 3. An equal split. A cone of height $H$ is to be cut by a plane parallel to the base so that the two parts have equal volume. Where must the cut be?

1. The small cone. It must have half the volume of the whole cone, so $f^3 = \dfrac{1}{2}$.
2. The fraction. $f = \dfrac{1}{\sqrt[3]{2}} = 0.794$.
3. The place. The small cone has height $0.794\,H$, so the cut is $0.206\,H$ above the base. The cut is not half way up. The frustum is wide and low.

Practise with new numbers

Try it in Python

This program finds how much metal and how much water a bucket needs, from its three measurements. Then it cuts cones at several heights and prints the volume ratios.

Common mistakes

• Using the wrong radius for the base of a bucket. If the wide end is open, the base is the small circle.
• Adding both circles for an open vessel. Count only the circles that are closed.
• Using the square for volumes. Volumes go with the cube of the scale factor, areas with the square.
• Writing the cut as half the volume when it is half the height. A cut half way up leaves $\frac{1}{8}$ of the volume on top.

Remember these

• Draw the side view, find $l$ if you need an area, and decide which surfaces count.
• A cone cut parallel to its base leaves a small cone similar to the whole one. Scale $f$ gives area $f^2$ and volume $f^3$.
• A cut half way up the height gives a frustum with $7$ times the volume and $3$ times the curved surface of the small cone.
• $1000\ \text{cm}^3$ is one litre.

Check yourself

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

1. A glass in the shape of a frustum has top radius $4\ \text{cm}$, bottom radius $3\ \text{cm}$ and height $10\ \text{cm}$. Find its volume in terms of $\pi$.

2. The small cone cut from the top of a cone has $\frac{1}{4}$ of the height of the whole cone. What fraction of the volume of the whole cone does it have?

3. For the same cut, what is the ratio of the curved surface of the frustum to that of the small cone?

4. A bucket is open at the top. Which circle is part of its surface, the larger or the smaller?

5. A cone is cut half way up its height. Is the volume of the small cone half the volume of the whole cone?