Cylinders

Volume of a cylinder = area of the circular base \times height.

First work out the area of the circular base, \pi r^2, in terms of \pi.

Use volume =\pi r^2h, and only round at the very end.

The formula uses the radius, which is half of the diameter.

Work out the volume in cubic centimetres first, then divide by 1000.

Volume of a cylinder =\pi r^2h, where r is the radius.

1. The radius is half the diameter: 10\div2=5 cm.
2. Volume =\pi r^2h=\pi\times5^2\times7.
3. \pi\times10^2\times7 uses the diameter instead of the radius, and 2\times\pi\times5\times7 is the area of the curved surface, not the volume.

Use volume =\pi r^2h with the lengths in metres.

1. Area of the base =\pi\times1.5^2=7.068...\text{ m}^2.
2. Volume =7.068...\times2.4=16.964...\text{ m}^3.
3. To the nearest whole number: 17\text{ m}^3.

Work out each volume in terms of \pi using \pi r^2h.

Part (a)

1. Volume of A =\pi\times4^2\times10=160\pi\text{ cm}^3.
2. Volume of B =\pi\times5^2\times6=150\pi\text{ cm}^3.
3. 160\pi is greater than 150\pi, so cylinder A has the greater volume.

Part (b)

1. 160\pi-150\pi=10\pi.
2. Cylinder A's volume is 10\pi\text{ cm}^3 greater.

Work out the area of the circular base in terms of \pi first.

1. Area of the base =\pi\times6^2=36\pi\text{ cm}^2.
2. Volume = area of base \times height, so 36\pi\times h=360\pi.
3. h=360\pi\div36\pi=10.

The plan is a circle: count squares to find its diameter. The front elevation shows the height.

1. The plan is a circle 4 squares across, so the diameter is 4 cm and the radius is 2 cm.
2. The front elevation is 6 squares tall, so the height is 6 cm.
3. Volume =\pi\times2^2\times6=24\pi\text{ cm}^3.

The water is itself a cylinder. Its height is the depth of the water, not the height of the vase.

1. The water is a cylinder with radius 6 cm and height 15 cm.
2. Volume of water =\pi\times6^2\times15=1696.46...\text{ cm}^3.
3. Correct to 1 decimal place: 1696.5\text{ cm}^3.

Work out the volume of the whole cylinder, then halve it. Remember to halve the diameter to get the radius.

1. Radius =40\div2=20 cm.
2. Volume of the whole cylinder =\pi\times20^2\times120=150796.4...\text{ cm}^3.
3. Half of this is 75398.2...\text{ cm}^3.
4. Correct to 3 significant figures: 75400\text{ cm}^3.

Divide the volume by the height to find the area of the base, \pi r^2.

1. Volume =\pi r^2\times5, so \pi r^2\times5=245\pi.
2. Divide both sides by 5\pi: r^2=49.
3. r=7, because 7\times7=49.

Work out the volume in cubic metres first, then multiply by 1000.

1. Volume =\pi\times0.4^2\times1.1=0.552920...\text{ m}^3.
2. In litres: 0.552920...\times1000=552.920... litres.
3. To the nearest litre: 553 litres.

Work out the volume of the pond in cubic metres, change it to litres, then divide by 15.

1. Volume =\pi\times1.2^2\times0.5=2.26194...\text{ m}^3.
2. In litres: 2.26194...\times1000=2261.94... litres.
3. Time =2261.94...\div15=150.79... minutes.
4. To the nearest minute: 151 minutes.

Work out the volume of a solid cylinder with the outside radius, then take away the volume of the hole.

1. Volume of the outside cylinder =\pi\times3^2\times40=1130.97...\text{ cm}^3.
2. Volume of the hole =\pi\times2.5^2\times40=785.39...\text{ cm}^3.
3. Volume of metal =1130.97...-785.39...=345.57...\text{ cm}^3.
4. Correct to 1 decimal place: 345.6\text{ cm}^3.

The volume of water stays the same. In cylinder B, depth = volume \div area of the base.

1. Volume of water =\pi\times6^2\times20=720\pi=2261.9...\text{ cm}^3.
2. Area of the base of B =\pi\times10^2=100\pi=314.15...\text{ cm}^2.
3. Depth =720\pi\div100\pi=7.2 cm.

Find the volume first. Then mass = density \times volume.

1. Volume =\pi\times4^2\times9=452.38...\text{ cm}^3.
2. Mass = density \times volume =0.9\times452.38...=407.15... g.
3. To the nearest gram: 407 g.

5 litres is 5000\text{ cm}^3. Work out the volume of each container in cubic centimetres.

1. 5 litres =5000\text{ cm}^3.
2. A: \pi\times10^2\times15=4712.3...\text{ cm}^3. This is less than 5000.
3. B: 18\times15\times20=5400\text{ cm}^3. This is more than 5000.
4. C: \pi\times9^2\times20=5089.3...\text{ cm}^3. This is more than 5000.
5. So B and C hold at least 5 litres.

Work out the volume of water in the jug and the volume of one glass. Divide, then think about whether to round up or down.

1. Volume of water =\pi\times7^2\times22=3386.6...\text{ cm}^3.
2. Volume of one glass =\pi\times3^2\times8=226.1...\text{ cm}^3.
3. 3386.6...\div226.1...=14.97...
4. There is not enough water for 15 full glasses, so the greatest number of full glasses is 14.