Volume of prisms

A fish tank 50 cm long, 30 cm wide and 40 cm deep holds 60,000 cm³ of water, which is exactly 60 litres.

How many 1-litre bottles fill the fish tank?

Try this before reading on. The tank in the lead is 50 cm long, 30 cm wide and 40 cm deep, and you are filling it from 1-litre bottles. How many bottles? Write down a guess first, then a calculation if you can. The answer is at the end of the next section. Getting there takes only two ideas: counting cubes, and knowing how big a litre is.

Layers of one-centimetre cubes

Volume answers one question: how many unit cubes fit inside? Picture a box 6 cm long, 4 cm wide and 3 cm high. The bottom layer holds 6 × 4 = 24 cubes of 1 cm. Three layers reach the top, so the box holds 24 × 3 = 72 cubes, or 72 cm³. Nobody counts cubes one by one. The picture is there because it explains the formula, volume = length × width × height, or V = lwh.

Now the litre. One litre is 1,000 cm³, the space inside a cube 10 cm on each side. The tank holds 50 × 30 × 40 = 60,000 cm³, and 60,000 ÷ 1,000 = 60. Sixty bottles. A cubic metre is a cube 100 cm on each side, so 1 m³ = 100 × 100 × 100 = 1,000,000 cm³ = 1,000 litres. There is a pattern here worth noticing. When you convert a length unit, a volume changes by the cube of the conversion factor, not by the factor itself: a hundred times the length means a million times the cubes.

One rule for every prism: base area times height

A prism is a solid with two identical, parallel ends (the bases) joined by flat rectangular sides. Slice it anywhere along its length and the cut looks the same, like slices from a loaf of bread, so each slice has the area of the base, B, and stacking h units of those slices gives V = B × h. When a prism lies on its side, the same measurement is usually called its length.

A ridge tent and a triangular chocolate box are both triangular prisms. The base is a triangle with base b and height t, so B = ½ b t and V = ½ b t L, where L is the length. The model starts with b = 4 cm, t = 3 cm and L = 6 cm, which gives V = ½ × 4 × 3 × 6 = 36 cm³.

Its surface is two triangles and three rectangles. The two triangles together make b × t, and the three rectangles are all L long, each as wide as one side of the triangle. Take a right triangle with legs of 3 cm and 4 cm; the third side is 5 cm. With a length of 10 cm the surface area is 2 × 6 + 10 × (3 + 4 + 5) = 12 + 120 = 132 cm². The model uses an isosceles triangle, so its two slanted sides match: s = √((b/2)² + t²).

Working backwards comes up just as often. A box must hold 120 cm³ and its base is 6 cm × 4 cm, so the height has to be 120 ÷ 24 = 5 cm. Builders, cooks and shippers do this every day. They know the volume they need and the floor space they have, and the height follows from the two.