Volume of Solids

La méthode, étape par étape

1. 1

   Identify the constant cross-section

   Triangular prism → the triangle is the cross-section

   A prism is any solid with the same shape all the way through. Finding that face is the whole method for prisms, and it makes cuboids, triangular prisms and cylinders a single case rather than three.

2. 2

   Volume of a prism: cross-section × length

   triangle area 30 cm², length 12 cm → 360 cm³

   Multiplying the face area by the length stacks that many layers of the cross-section. Three lengths have been multiplied, so the unit is cm³ — the cube is again a consequence of the arithmetic.

3. 3

   Cylinder: πr²h

   r = 4 cm, h = 10 cm → π × 16 × 10 = 502.7 cm³

   A cylinder is a prism with a circular cross-section, so its volume is the circle's area times the height. Nothing new is being learned here, which is worth pointing out — the formula is πr² with an h attached.

4. 4

   Cone and pyramid: a third of the prism

   cone = ⅓πr²h · pyramid = ⅓ × base area × h

   A cone occupies exactly a third of the cylinder that encloses it, and a pyramid a third of its prism. Remembering the one-third rather than two separate formulas reduces this to a single fact.

5. 5

   Sphere: 4/3 πr³

   r = 3 cm → 4/3 × π × 27 = 113.1 cm³

   The sphere formula has to be memorised; it does not follow from anything else at this level. Note the cube on the r — doubling a sphere's radius multiplies its volume by eight, not by two.

Every prism is the same formula

Cuboids, triangular prisms, hexagonal prisms and cylinders are all handled by cross-section times length. Students who learn a separate formula for each have four things to remember and no way to cope with a fifth shape; students who learn the principle can handle any prism they are shown.

The test for a prism is whether slicing it anywhere along its length gives the same shape. A cone fails that test, which is why it needs its own formula — and why the one-third relationship is worth knowing rather than deriving.

• Prism: cross-section area × length
• Cylinder: πr²h
• Cone: ⅓πr²h — a third of its cylinder
• Pyramid: ⅓ × base area × height
• Sphere: 4/3 πr³

Converting volume units

There are 100 cm in a metre, but there are 1,000,000 cm³ in a cubic metre — because the conversion applies in three dimensions and 100³ is a million. Multiplying by 100 instead is wrong by a factor of ten thousand.

The same catches area: 10,000 cm² in a square metre, not 100. Working out the conversion factor from the power rather than recalling it means never having to remember which large number belongs to which.

Volume and surface area answer different questions

Volume is how much fits inside, in cm³. Surface area is how much material covers the outside, in cm². Filling a tank is volume; painting it or wrapping it is surface area, and questions are usually phrased with one of those verbs.

The units check applies here as it did with perimeter and area. An answer in cm³ to a question about how much wrapping paper is needed has answered the wrong question.

How we teach volume

We teach prisms as one idea and only then introduce the cone, pyramid and sphere as the exceptions. Presenting five formulas at once makes the topic look like memorisation, when three of the five are the same formula wearing different shapes.

We drill the cubic unit conversion separately, because it is counter-intuitive and it appears in questions about tanks, pools and packaging where the marks are otherwise straightforward.