Given three sides, there is exactly one triangle, and a pair of compasses finds its third corner for you. This lesson covers measuring, constructing triangles with arcs left showing, and using nets to find surface areas and volumes.
To construct a triangle from three sides: draw one side with a ruler, then draw an arc from each end with the compasses set to the other two lengths. The arcs cross at the third vertex. Leave the arcs showing.
Fig. 1: triangle ABC with AB = 8 cm, AC = 6 cm and BC = 5 cm, drawn to scale. The two construction arcs cross at C
Section 01
Measuring and drawing.
Every construction starts with accurate measuring. Lengths are measured to the nearest millimetre with a ruler and angles to the nearest degree with a protractor.
Lines: start at the 0 mark, not the end of the ruler. Use a sharp pencil; a thick line hides a millimetre.
Angles: put the protractor's centre cross exactly on the vertex and its zero line along one arm. Then count from the 0 on THAT arm; this tells you whether to read the inner or the outer scale.
Sense check: decide first whether the angle is acute or obtuse. If it looks acute but you read 130°, you used the wrong scale.
Reflex angles: a protractor only goes to 180°. Measure the angle on the other side and subtract it from 360°.
Worked example Drawing a reflex angle
Explain how to draw an angle of \(250^\circ\) with a 180° protractor.
The reflex angle and the angle on the other side make a full turn: \(360^\circ - 250^\circ = 110^\circ\).
Draw a line, and at one end measure and draw an angle of \(110^\circ\).
The reflex angle is the angle OUTSIDE this \(110^\circ\) angle; mark it with an arc and label it \(250^\circ\).
Answer: Draw \(110^\circ\) and mark the outside angle as \(250^\circ\).
Examiner trap
Reading the wrong scale on the protractor is the most common slip: \(50^\circ\) read as \(130^\circ\). Always decide acute or obtuse by eye before you read the number.
Section 02
A triangle from three sides.
In the lab, set three side lengths and build the triangle step by step: draw the base, swing an arc from each end, and see where they cross. Then try lengths 3, 4 and 8 and watch the arcs fail to meet.
Step 1: draw the longest side accurately with a ruler and label its ends.
Step 2: set the compasses to the second length, put the point on one end, and draw an arc above the line.
Step 3: set the compasses to the third length, put the point on the OTHER end, and draw an arc that crosses the first one.
Step 4: join the crossing point to both ends with a ruler. Do not rub out the arcs: they earn the method mark.
When it is impossible: the two shorter sides must add up to MORE than the longest side, otherwise the arcs never meet.
Worked example Construct a triangle
Describe how to construct triangle ABC with AB = 8 cm, AC = 6 cm and BC = 5 cm (Fig. 1). Check first that it is possible.
Check: the two shorter sides give \(6 + 5 = 11\) cm, which is more than 8 cm, so the triangle exists.
Draw AB = 8 cm with a ruler.
With compasses set to 6 cm and the point on A, draw an arc above AB.
With compasses set to 5 cm and the point on B, draw an arc to cross the first arc. The crossing point is C.
Join AC and BC with a ruler, leaving both arcs visible.
Answer: Triangle ABC as in Fig. 1, with both construction arcs showing.
Worked example A rhombus from two triangles
A rhombus PQRS has sides of 5 cm and diagonal PR = 8 cm. Explain how to construct it, and find the length of the other diagonal QS.
Draw PR = 8 cm.
Set the compasses to 5 cm. Draw arcs from P and from R above PR; they cross at Q.
Draw arcs from P and R with the same radius BELOW PR; they cross at S. Join up PQRS.
The diagonals of a rhombus bisect each other at 90°, so each half of QS is \(\sqrt{5^2 - 4^2} = \sqrt{9} = 3\) cm.
So QS = \(2 \times 3 = 6\) cm, which you can check by measuring.
Answer: Two triangles on PR; QS = 6 cm
Your turn
Can a triangle be constructed with sides 4 cm, 5 cm and 10 cm? Explain.
No. The two shorter sides give \(4 + 5 = 9\) cm, which is less than 10 cm, so the arcs from the ends of the 10 cm line never meet.
Examiner trap
Rubbing out the construction arcs loses marks even when the triangle is perfect. Cambridge says construction arcs must be shown. Also, set each arc's radius from the correct end: the 6 cm arc goes round A because AC = 6 cm.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
To draw an angle of \(300^\circ\) with a 180° protractor, which angle do you actually measure?
Quick check+10 XP
When constructing a triangle from three sides, what happens to the construction arcs?
Quick check+10 XP
Which three lengths can form a triangle?
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How many rectangles are in the net of a pentagonal prism?
Section 03
Nets, surface area and volume.
Fold and unfold solids in the lab. Pick a cuboid and a triangular prism, open each into a net, and count the rectangles of each size. The total area of the net is the surface area of the solid.
Cuboid \(l \times w \times h\): the net has three pairs of rectangles, so surface area \(= 2(lw + lh + wh)\) and volume \(= lwh\).
Prism: two congruent end faces plus one rectangle for every side of the cross-section. Volume = area of cross-section \(\times\) length.
Pyramid: the base plus one triangle for each side of the base, all meeting at the apex when folded.
Drawing a net: use a ruler, measure every edge accurately, and make edges that will join the same length.
Worked example From a net to surface area and volume
The net of a triangular prism has two right-angled triangles with sides 3 cm, 4 cm and 5 cm, and three rectangles each 10 cm long. Find the surface area and the volume of the prism.
Each triangle has area \(\tfrac{1}{2} \times 3 \times 4 = 6\) cm2; two of them give 12 cm2.
The rectangles are \(3 \times 10\), \(4 \times 10\) and \(5 \times 10\): total \((3 + 4 + 5) \times 10 = 120\) cm2.
Do not forget the factor of 2 for the pairs of faces in a cuboid, and do not include the slanted 5 cm side in the triangle's area: the area of a right-angled triangle uses the two sides that meet at the right angle.
Stage 2 · Exam
Exam-style questions
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Which set of side lengths can NOT be used to construct a triangle?
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Find the total surface area of a cuboid that measures 5 cm by 4 cm by 2 cm.
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The net of a pyramid is a square of side 6 cm with four triangles, each with base 6 cm and height 5 cm. Find the surface area of the pyramid.
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A rhombus ABCD has sides of 7 cm and diagonal AC = 10 cm. After drawing AC, what is the next step?
Skill unlocked
Arcs drawn, triangles constructed.
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