Cylinders

A cylinder is a solid with two identical circular ends and a curved side, like a tin of beans. In the exam, the volume of a cylinder usually comes near the end of the paper, as the first part of a longer question worth 3 or 4 marks.

Paper 1 has no calculator, so there the answer is usually given in terms of \pi, sometimes from a plan and front elevation drawn on a grid. On Papers 2 and 3 you use the \pi key and round only at the end, for example correct to 1 decimal place.

The longer questions ask things like how many minutes a tank takes to fill, or how many litres it holds. Some use the volume with a density, to find a mass or the density itself. Questions about painting the outside of a cylinder need its surface area instead, which has its own page.

Since 2022 the exam has come with a formulae sheet. It gives the area of a circle and the volume of a prism, but not the volume of a cylinder, so learn the formula on this page.

A cylinder is a prism. Cut straight across it anywhere and you get the same circle, so the circle is its cross-section. The volume of any prism is the area of the cross-section \times the length.

The circular end has radius r. The height h is the distance between the two ends. For a cylinder lying on its side, it is called the length.

The cross-section is a circle with area \pi r^2, and the length of the prism is the height. So:

V=\pi r^2h

where V is the volume, r is the radius and h is the height.

Think of the cylinder as layers of the circle, each 1 cm thick. A cylinder 5 cm tall is 5 layers. Volume is measured in cubic units, such as \text{cm}^3 or \text{m}^3

The formula needs the radius. If you are given the diameter, halve it first: a diameter of 10 cm means a radius of 5 cm. Putting the diameter in by mistake makes the volume four times too big, because the radius is squared.

On Paper 1 you have no calculator, so you leave \pi in the answer. This is called giving your answer in terms of \pi.

Work out r^2 \times h as an ordinary number, then write \pi after it. Square the radius first. A cylinder with radius 3 cm and height 7 cm has 3^2 \times 7 = 9 \times 7 = 63, so its volume is 63\pi\text{ cm}^3

A plan and front elevation show the same thing. The plan is 8 squares across, so the diameter is 8 cm and the radius is 4 cm. The front elevation is 3 squares tall, so the height is 3 cm. The volume is \pi \times 4^2 \times 3 = 48\pi\text{ cm}^3

Some questions give you the volume and ask for the height. Put what you know into V = \pi r^2 h, then undo the multiplying by dividing. In the same way, the area of the circular end is the volume \div the height.

With a calculator, find the area of the circle first, then divide the volume by it. A cylinder with radius 5 cm and volume 500\text{ cm}^3 has an end of area \pi \times 5^2 = 78.539\ldots\text{ cm}^2, so its height is 500 \div 78.539\ldots = 6.4 cm, correct to 1 decimal place.

To find the radius instead, divide the volume by \pi and by the height to get r^2, then find the square root. A cylinder with height 3 cm and volume 108\pi\text{ cm}^3 has r^2 = 108 \div 3 = 36, so r = 6 cm.

Some questions say what a solid cylinder is made from and give the density of that material, the mass of each 1\text{ cm}^3 of it. Density has its own page; for a cylinder, this is the formula you need:

\text{density} = \frac{\text{mass}}{\text{volume}}

So mass = density \times volume. The formula is not on the formulae sheet and the question does not give it, so learn it.

For a cylinder, work out the volume first with V = \pi r^2 h, then use the density. Keep the full calculator value of the volume and round only at the end.

Capacity is how much a container holds. For liquids, use these facts:

1\text{ litre} = 1000\text{ cm}^3 \qquad 1\text{ m}^3 = 1000\text{ litres}

A jug with volume 1500\text{ cm}^3 holds 1500 \div 1000 = 1.5 litres. A tank with volume 2\text{ m}^3 holds 2 \times 1000 = 2000 litres.

If a container is only part full, use the depth of the liquid as the height. A vase with radius 4 cm and water 10 cm deep holds \pi \times 4^2 \times 10 = 160\pi\text{ cm}^3 of water, however tall the vase is.

To find how long a container takes to fill, divide the amount by the rate. Change the volume into the unit of the rate first, such as litres for a rate in litres per minute. Round as the question says; for a number of glasses filled completely, round down.

• Using the diameter as the radius. Halve the diameter first; only the radius goes into \pi r^2 h
• Stopping at the area or the circumference of the circle. Multiply the area of the circle by the height to get the volume.
• Multiplying the radius or the diameter by the height. That is not a volume; use V = \pi r^2 h
• Multiplying the area of the plan by the area of the front elevation. Read the radius from the plan and the height from the elevation, then use \pi r^2 h
• Leaving the volume in the wrong units. Change \text{cm}^3 or \text{m}^3 into litres before you use a rate in litres per minute.
• Multiplying by the rate instead of dividing. To find a time, divide the amount by the rate.