Volume of Solids
Senior 2 · Mathematics
Volume of Solids
Lesson 1 of 4
22 min
How much water does a school's rainwater tank hold? How many bags of cement fill a foundation? How much air is in a classroom? Each question asks for a volume: the amount of space a solid takes up. Volume is measured in cubic units such as cm3 and m3, and liquid capacity in litres.
This lesson starts with the simplest solids. Every later formula in the unit grows from one idea you will meet here: volume = area of the base × height.
A face is a flat surface, an edge is where two faces meet, and a vertex (plural vertices) is a corner. A cuboid has 6 faces, 12 edges and 8 vertices.
A prism is a solid with two identical, parallel end faces joined by rectangles. If you slice it parallel to an end, every slice (the cross-section) is the same shape and size. A cuboid is a prism with a rectangular cross-section; a triangular prism, like the roof of a house, has a triangular cross-section with 5 faces, 9 edges and 6 vertices.
A cuboid of length l, width w and height h holds l × w layers-worth of unit cubes in each layer, and h layers:
V = l × w × h
A cube has all edges equal to a, so V = a3.
Worked example 1
A classroom is 8 m long, 6 m wide and 3 m high. Its volume is 8 × 6 × 3 = 144 m3 of air.
Worked example 2
A cube has a volume of 64 cm3. Its edge is the cube root: a = ∛64 = 4 cm.
In the cuboid, l × w is the area of the base. The same idea works for every prism:
V = area of cross-section × length (or base area × height).
Worked example 3
A tent is a triangular prism. Its triangular end has a base of 3 m and a height of 2 m, and the tent is 5 m long.
Area of the triangle = ½ × 3 × 2 = 3 m2. Volume = 3 × 5 = 15 m3.
Worked example 4
A concrete water tank is 2 m long, 1.5 m wide and 1.2 m deep. V = 2 × 1.5 × 1.2 = 3.6 m3 = 3,600 litres.
Tip
Exam technique. Convert every length to the same unit before you multiply: mixing 2 m with 50 cm is the fastest way to lose the mark. For a prism, first find the area of the end face (triangle, trapezium, …), then multiply by the length. And remember that 1 m3 is 1000 litres, not 100.
Find the volume of a brick 20 cm long, 10 cm wide and 6 cm high.
Solution: 20 × 10 × 6 = 1,200 cm3.
Find the volume of a cube of edge 5 cm.
Solution: 53 = 125 cm3.
A prism has a trapezium cross-section with parallel sides 4 m and 6 m, 3 m apart. The prism is 10 m long. Find its volume.
Solution: Area of trapezium = ½ × (4 + 6) × 3 = 15 m2. Volume = 15 × 10 = 150 m3.
How many litres does a tank 2 m by 1 m by 1 m hold?
Solution: 2 m3 = 2,000 litres.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
