4.10Foundation tier1MA1 · Section 4: Geometry and trigonometry

Volume & surface area (prisms, cylinders, spheres, cones, pyramids)

Key learnings

Volume formulae
Prism $=$ cross-section area $\times$ length; cylinder $\pi r^2 h$ ; sphere $\tfrac43\pi r^3$ ; cone $\tfrac13\pi r^2 h$ ; pyramid $\tfrac13\times$ base area $\times$ height.
Surface area = add up every face
For a prism, add the two ends and all the rectangular faces; for a cylinder, $2\pi r^2 + 2\pi r h$ ; cone curved surface $\pi r l$ (slant height $l$ ).

Common mistakes

Drawn from real examiner reports.

Hemisphere: using the sphere volume without halving

Forgetting to divide the sphere formula by 2 for a hemisphere, or using $r^2$ instead of $r^3$ .

Surface area: missing or doubling faces

Finding the volume by mistake, not halving the triangle area, forgetting the slant face, or forgetting one triangular end.

Wrong volume–capacity conversion

Not knowing $1000\ \text{cm}^3 = 1$ litre (and $1\ \text{m}^3 = 1000$ litres); dividing by 100 or 10000.

Exam tips

Use brackets when substituting
Put brackets around a substituted expression (e.g. $(5x)^2$ ) so the whole thing is squared.

Revision notes

Solid	Volume
Prism	cross-section area $\times$ length
Cylinder	$\pi r^2 h$
Sphere	$\tfrac43 \pi r^3$
Cone	$\tfrac13 \pi r^2 h$
Pyramid	$\tfrac13 \times$ base area $\times$ height

A hemisphere is half a sphere: $\tfrac12 \times \tfrac43\pi r^3 = \tfrac23\pi r^3$ . (Use $r^3$ , not $r^2$ .)

Flashcards

Foundation

Volume of a cylinder?

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Practice questions

Question 1 of 3Core4 marks

The diagram shows a solid prism. Its cross-section is a right-angled triangle with sides $6$ cm, $8$ cm and $10$ cm (the $6$ cm and $8$ cm sides meet at the right angle). The prism is $20$ cm long.

Work out the total surface area of the prism.

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