Construction of Geometric Figures

Using only a compass and ruler to bisect a line, bisect an angle, construct 60° and 90°, and build a triangle from given sides — the constructions that appear as their own exam question, with step-by-step diagrams.

"Construct" has a precise meaning in maths: build the figure using only a compass and a straight edge. Measuring the answer with a protractor scores nothing — the marks are for the arcs, which show how you got there. Never rub them out.

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Golden rule: leave all your construction arcs visible. They are the working, and they carry the marks.

1Bisecting a line segment (perpendicular bisector)

Equal arcs swung from A and from B cross above and below the line. Joining those crossings cuts AB in half at 90°.

Worked ExampleConstruct the perpendicular bisector of AB

1
Open the compass to more than half of AB.
If it is less than half, the arcs will never meet.
2
With the point on A, draw arcs above and below the line.
Keep the compass setting unchanged.
3
With the same setting and the point on B, draw two more arcs.
They cut the first pair at two points.
4
Rule a line through the two crossing points.
This line bisects AB and is perpendicular to it.

2Bisecting an angle

An arc from the vertex marks P and Q on the two arms. Equal arcs from P and Q then cross on the bisector.

Worked ExampleBisect the angle at V

1
Put the compass point on V and draw an arc cutting both arms, at P and Q.
P and Q are now the same distance from V.
2
From P, draw an arc inside the angle. From Q, with the same setting, draw another.
The two arcs cross at one point.
3
Rule a line from V through that crossing point.
It splits the angle into two equal halves.

3Constructing 60° and 90°

60° — draw an arc from the vertex; then, keeping the same radius, put the compass on where that arc met the line and swing again. Joining the vertex to the new crossing gives exactly 60° (it is the corner of an equilateral triangle).
90° — construct the perpendicular bisector of a line, as above.
30° — construct 60°, then bisect it. 45° — construct 90°, then bisect it.

4Constructing a triangle from three sides

Draw the longest side first, then swing an arc of each remaining length from each end. Where they cross is the third vertex.

Worked ExampleConstruct △ABC with AB = 7 cm, AC = 5 cm and BC = 6 cm

1
Rule AB = 7 cm with a ruler.
Start with the longest side; it keeps the figure on the page.
2
Set the compass to 5 cm and swing an arc from A.
Every point on that arc is 5 cm from A.
3
Set the compass to 6 cm and swing an arc from B.
Every point on it is 6 cm from B.
4
Where the arcs cross is C. Join AC and BC.
C is the only point 5 cm from A and 6 cm from B.

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If two sides added together are shorter than the third, the arcs never meet and the triangle cannot exist. Check that the two shorter sides add to more than the longest one before you start.

5Investigating properties from your construction

Exams often follow a construction with "now measure…" — for example, construct an equilateral triangle and measure its angles (each is 60°), or construct an isosceles triangle and check that the base angles are equal.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
What two instruments may you use in a construction?
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A compass and a ruler (straight edge) only.
2
Why must you not rub out your arcs?
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The arcs are the working — the marks are awarded for them.
3
When bisecting AB, how wide must the compass be opened?
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More than half of AB, or the arcs will not cross.
4
What angle does a perpendicular bisector make with the line?
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90°
5
How do you construct a 30° angle?
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Construct 60°, then bisect it.
6
How do you construct a 45° angle?
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Construct 90°, then bisect it.
7
Which side should you draw first when constructing a triangle?
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The longest side.
8
Can a triangle with sides 3 cm, 4 cm and 9 cm be constructed?
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No — 3 + 4 = 7, which is less than 9, so the arcs never meet.
9
Each angle of a constructed equilateral triangle measures…
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60°