Lesson Guide
Explanation
Volume is the amount of space inside a solid, measured in cubic units. Think of this box as layers. The base is six by three, so one layer one centimetre thick holds eighteen centimetre cubes. The box is four layers high, so the volume is eighteen times four, which is seventy two cubic centimetres. This works for every prism: find the area of the cross-section, the face that is the same all the way through, and multiply by how far it runs. V equals A h.
Your Turn
Find the volume of a box 5 cm × 5 cm × 2 cm. Check: A = 25, so V = 25 × 2 = 50 cm3.
Watch Out
Volume is in cm3. The base area 18 is in cm2, and it only becomes a volume once you multiply by the height.
Explanation
For any other prism, find the area of the cross-section first. The top prism has a right-angled triangle with sides six and four, so its area is one half times six times four, twelve square centimetres. It runs for nine centimetres, so the volume is twelve times nine, one hundred and eight cubic centimetres. The bottom prism has a trapezium with parallel sides five and nine and height four. Its area is one half times five plus nine, times four, which is twenty eight. It runs for ten, so its volume is two hundred and eighty cubic centimetres.
Your Turn
A triangle with base 10 cm and height 3 cm runs for 4 cm. Check: A = 15, so V = 60 cm3.
Watch Out
Keep the ½. 6 × 4 × 9 = 216 is the volume of a box, twice the triangular prism.
Explanation
A cylinder is a prism whose cross-section is a circle, so the same rule works. The area of the circle is pi r squared. With a radius of three, that is nine pi square centimetres. Multiply by the height, seven: the volume is sixty three pi cubic centimetres. Leave pi in the answer. It is exact, and it needs no calculator.
Your Turn
Find the volume of a cylinder with r = 2 cm and h = 5 cm. Check: 4π × 5 = 20π cm3.
Watch Out
2πrh is the curved surface AREA, not the volume. For r = 3, h = 7 it gives 42π, which is not 63π.
Explanation
Capacity is how much a container holds. One cubic centimetre holds exactly one millilitre, and one thousand millilitres make a litre. This box is twenty by ten by fifteen, so its volume is three thousand cubic centimetres. That is three thousand millilitres, which is three litres. A useful picture: a ten centimetre cube holds exactly one litre. And a cubic metre is a hundred by a hundred by a hundred centimetres, a million cubic centimetres, which is one thousand litres.
Your Turn
A box 25 cm × 8 cm × 10 cm. Check: 2000 cm3 = 2000 mL = 2 L.
Watch Out
1 m3 is 1000 L, not 100 L: a metre is 100 cm, but a cubic metre is 100 × 100 × 100 cm3.
Explanation
Sometimes you know the volume and need a length. This cylinder has radius five and volume two hundred pi. Its cross-section is pi times five squared, twenty five pi. So twenty five pi times h equals two hundred pi. Divide both sides by twenty five pi: h is eight centimetres. The same works for any prism. If the cross-section is fifteen square centimetres and the volume is ninety cubic centimetres, the length is ninety divided by fifteen, which is six centimetres. Quiz and exercises now.
Your Turn
A box has a 6 cm × 4 cm base and volume 120 cm3. Find its height. Check: 120 ÷ 24 = 5 cm.
Watch Out
Divide by the whole cross-section 25π, not by the radius 5. 200 ÷ 5 = 40 is not the height.
📌 Key Points — this lesson in a nutshell
- Every prism: V = Ah, the area of the cross-section × the length (or height) it runs for. Units are cubed: cm3, m3.
- A cylinder is a prism with a circle for its cross-section: V = πr2h. Halve a diameter first, and leave the answer in terms of π.
- 1 cm3 = 1 mL, 1000 cm3 = 1 L, and 1 m3 = 1000 L.
- Missing length: divide the volume by the cross-section area, h = V ÷ A. Missing radius: r2 = V ÷ (πh), then take the square root.
Step 1: Introduction
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