GCSE Maths revision · Geometry and measures

Cones, spheres and pyramids

Foundation and Higher

The idea

Volume of a pyramid or cone is $\tfrac{1}{3} \times$ base area $\times$ height. A sphere has $V = \tfrac{4}{3}\pi r^3$ and surface area $4\pi r^2$ . The exam gives you these formulae when you need them.

Identify the radius and the perpendicular height.
Substitute into the formula given in the question.
For a composite solid, find each part and add.

Watch out Halve the diameter first: the formulae all use the radius.

See it worked

Worked exampleCalculator allowed2 marks

A sphere has a radius of 6 cm. Volume of a sphere $= \frac{4}{3}\pi r^3$ , where $r$ is the radius.

Work out the volume of the sphere. Give your answer correct to 3 significant figures.

Substitute $r = 6$ . $\tfrac{4}{3} \times \pi \times 6^3 = 288\pi = 904.77\ldots$
Correct to 3 significant figures: $905\text{ cm}^3$ .

Answer $905\text{ cm}^3$

Your turn

Question 1Non-calculator2 marks

A pyramid has a square base of side 6 cm. Its perpendicular height is 10 cm and each sloping edge is 11.6 cm long. Volume of a pyramid $= \frac{1}{3} \times$ area of base $\times$ perpendicular height.

Work out the volume of the pyramid.

Question 2Calculator allowed5 marks

A solid cone has base radius 5 cm, perpendicular height 12 cm and slant height 13 cm. Curved surface area of a cone $= \pi r l$ , where $l$ is the slant height. Volume of a cone $= \frac{1}{3}\pi r^2 h$ .

(a) Work out the volume of the cone. Give your answer correct to 3 significant figures.

(b) Work out the total surface area of the cone. Give your answer correct to 3 significant figures.

Question 3Non-calculator4 marks

A paper sector has radius 15 cm and angle $216^\circ.$ Its two straight edges are joined without overlap to form the curved surface of a cone. Work out the volume enclosed by the cone. Give your answer in terms of $\pi .$

Volume of a cone $= \frac{1}{3}\pi r^2 h$

You must show your working.

In past papers

62 official questions in 60 papers since 2017. The latest:

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