IGCSE · Practice questions · Ruler and compasses constructions

Build it accurately.

Six original Cambridge-style questions on constructions: measuring, constructing triangles with ruler and compasses, and using nets to find surface area and volume.

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

Line, arc, arc, join.

Check first: the two shorter sides must add to more than the longest.
Base: draw one side accurately with a ruler.
Arcs: compasses set to each other length, centred on the matching end.
Join and keep: join to the crossing point; leave the arcs showing.
Nets: surface area = total area of every face in the net.

Recall

[3 marks]

Describe, step by step, how to construct triangle PQR with PQ = 7 cm, PR = 5 cm and QR = 4 cm using a ruler and compasses only.

Draw PQ = 7 cm with a ruler. ✓
Compasses set to 5 cm, point on P: draw an arc above PQ. ✓
Compasses set to 4 cm, point on Q: draw an arc crossing the first arc at R. ✓
Join PR and QR with a ruler; leave the arcs showing. ✓

Analysis

[2 marks]

For each set of lengths, say whether a triangle can be constructed. Give a reason.

(a) 2 cm, 5 cm, 8 cm (b) 5 cm, 6 cm, 10 cm

(a) No: \(2 + 5 = 7 < 8\), so the arcs do not meet. ✓
(b) Yes: \(5 + 6 = 11 > 10\). ✓

Calculation

[1 marks]

Kofi wants to draw an angle of \(290^\circ\) using a protractor that goes up to \(180^\circ\). What angle should he measure?

\(360^\circ - 290^\circ = 70^\circ\). He draws \(70^\circ\) and marks the reflex angle outside it. ✓

Calculation

[3 marks]

The net of a cube has a total area of 150 cm². Find the volume of the cube.

A cube's net has 6 equal squares: each has area \(150 \div 6 = 25\) cm². ✓
Edge length \(= \sqrt{25} = 5\) cm. ✓
Volume \(= 5^3 = 125\) cm³. ✓

Calculation

[4 marks]

A triangular prism is 15 cm long. Its cross-section is a right-angled triangle with sides 6 cm, 8 cm and 10 cm.

(a) Sketch the net and list the faces. (b) Find the surface area. (c) Find the volume.

(a) Two right-angled triangles (6, 8, 10) and three rectangles: \(6 \times 15\), \(8 \times 15\), \(10 \times 15\). ✓
(b) Triangles: \(2 \times \tfrac{1}{2} \times 6 \times 8 = 48\). Rectangles: \((6 + 8 + 10) \times 15 = 360\). Total 408 cm². ✓
(c) \(24 \times 15 = 360\) cm³. ✓

Analysis

[4 marks]

A box with no lid is made from a net. The box is a cuboid 20 cm long, 12 cm wide and 8 cm high.

(a) How many faces does the net have? (b) Find the area of card in the net. (c) Will the box hold 2 litres of rice? (1 litre = 1000 cm³)

(a) A cuboid has 6 faces; with no lid the net has 5. ✓
(b) Base \(20 \times 12 = 240\); two long sides \(2 \times 20 \times 8 = 320\); two short sides \(2 \times 12 \times 8 = 192\). ✓
Total \(240 + 320 + 192 = 752\) cm². ✓
(c) Volume \(= 20 \times 12 \times 8 = 1920\) cm³ = 1.92 litres, which is less than 2 litres, so no. ✓

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.