IGCSE · Practice questions · Surface area and volume: cones, pyramids and spheres

The right formula, the right length.

Six original Cambridge-style questions on the volume and surface area of pyramids, cones, spheres and hemispheres, using the List of formulas.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Pick the formula, then the length it needs.

Pyramid and cone volume: \(\tfrac{1}{3}\) of base area \(\times\) vertical height.
Cone curved surface: \(\pi rl\) with slant height \(l = \sqrt{r^2 + h^2}\).
Sphere: \(V = \tfrac{4}{3}\pi r^3\), \(A = 4\pi r^2\).
Solid shapes: add flat faces (cone base \(\pi r^2\), hemisphere face \(\pi r^2\)).
Melting and recasting: the volume stays the same.

Recall

[2 marks]

A pyramid has a square base of side 8 cm and vertical height 9 cm. Find its volume.

Base area \(= 64\text{ cm}^2\). ✓
\(V = \tfrac{1}{3} \times 64 \times 9 = 192\text{ cm}^3\) ✓

Calculation

[2 marks]

A sphere has diameter 14 cm. Calculate (a) its volume (b) its surface area.

\(r = 7\) cm. ✓
(a) \(\tfrac{4}{3}\pi \times 343 = 1436.7... = 1440\text{ cm}^3\) (3 s.f.) ✓
(b) \(4\pi \times 49 = 615.7... = 616\text{ cm}^2\) ✓

Calculation

[3 marks]

A cone has radius 4 cm and vertical height 9 cm. Calculate (a) its volume (b) its curved surface area.

(a) \(\tfrac{1}{3}\pi \times 16 \times 9 = 48\pi = 150.7... = 151\text{ cm}^3\) ✓
(b) \(l = \sqrt{16 + 81} = \sqrt{97} = 9.848...\) cm ✓
Curved surface \(= \pi \times 4 \times 9.848... = 123.7... = 124\text{ cm}^2\) ✓

Calculation

[2 marks]

Without a calculator, find the volume and the total surface area of a solid hemisphere of radius 3 cm. Give your answers in terms of \(\pi\).

\(V = \tfrac{1}{2} \times \tfrac{4}{3}\pi \times 27 = 18\pi\text{ cm}^3\) ✓
Surface \(= \tfrac{1}{2} \times 4\pi \times 9 + \pi \times 9 = 18\pi + 9\pi = 27\pi\text{ cm}^2\) ✓

Analysis

[3 marks]

A ball is a sphere with volume \(500\text{ cm}^3\).

Calculate (a) its radius (b) its surface area.

(a) \(\tfrac{4}{3}\pi r^3 = 500\), so \(r^3 = \dfrac{1500}{4\pi} = 119.3...\) and \(r = 4.923... = 4.92\) cm ✓
(b) \(4\pi \times 4.923...^2 = 304.6... = 305\text{ cm}^2\) (use the unrounded radius) ✓

Analysis

[4 marks]

A solid metal sphere of radius 6 cm is melted down. All the metal is used to make solid cones, each with radius 2 cm and vertical height 3 cm.

How many cones are made?

Sphere: \(\tfrac{4}{3}\pi \times 216 = 288\pi\text{ cm}^3\). ✓
One cone: \(\tfrac{1}{3}\pi \times 4 \times 3 = 4\pi\text{ cm}^3\). ✓
Number of cones \(= 288\pi \div 4\pi = 72\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.