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Last reviewed: 12 November 2026
Updated November 2026. These Grade 7 geometrical constructions notes explain how to bisect an angle and how to construct 60°, 30°, 90°, 45°, 120° and 105° using only a ruler and a pair of compasses, then how to construct triangles and circles. Quick answer: 60° comes from one arc stepped along a line; bisect it to get 30°. Two arcs stepped along give 120°, and bisecting the angle between 60° and 120° gives 90°. Bisect 90° for 45°.
These are original revision notes for Kenyan junior school learners. The skills follow the Geometrical Constructions sub-strand of Geometry in the Grade 7 Mathematics design on the KICD curriculum designs page. Measuring and naming angles is covered in the Grade 7 mathematics revision guide, and all notes are listed on the Grade 7 notes hub.
On this page
What You Need and the Golden Rules
You need a sharp pencil, a ruler and a pair of compasses. A protractor is only used to check your work, never to draw the angle. Keep the compasses at the same radius throughout a step unless told otherwise, and leave all arcs visible: in an exam, the arcs show the examiner that you constructed the angle.
Bisecting an Angle
To bisect means to cut into two equal parts.
- Draw angle ABC with vertex B.
- With B as centre, draw an arc that cuts BA at P and BC at Q.
- With P as centre and any radius more than half of PQ, draw an arc inside the angle. Keep the same radius and repeat with Q as centre, so the two arcs cross at R.
- Join B to R. BR bisects the angle, so angle ABR = angle RBC.
Check with a protractor: if ABC was 80°, each half should measure 40°.
Constructing 60° and 120°
- Draw a line and mark a point A on it.
- With A as centre, draw a large arc cutting the line at P.
- With P as centre and the same radius, draw an arc cutting the first arc at Q. Join A to Q. Angle QAP = 60°.
- With Q as centre and the same radius, cut the first arc again at R. Join A to R. Angle RAP = 120°.
This works because A, P and Q form an equilateral triangle, and every angle of an equilateral triangle is 60°. OpenStax Prealgebra’s section on angles and triangles gives the angle facts these constructions rely on.
Constructing 30°, 90°, 45°, 105° and Multiples of 7.5°
- 30°: construct 60°, then bisect it.
- 90°: construct 60° and 120° from the same point, then bisect the angle between them (60° + 30° = 90°).
- 45°: construct 90°, then bisect it.
- 15°: bisect 30°. 7.5°: bisect 15°.
- 105°: construct 90° and 120°, then bisect the angle between them (90° + 15° = 105°).
- 75°: bisect the angle between 60° and 90° (60° + 15°).
Any angle that is a multiple of 7.5°, such as 22.5° or 67.5°, can be built from these steps by adding or bisecting.
Constructing Triangles
Given three sides
Construct triangle XYZ with XY = 7 cm, YZ = 5 cm and XZ = 6 cm. Draw XY = 7 cm. With X as centre and radius 6 cm, draw an arc. With Y as centre and radius 5 cm, draw an arc cutting the first at Z. Join XZ and YZ.
Given two sides and the angle between them
Construct triangle ABC with AB = 6 cm, angle BAC = 60° and AC = 4.5 cm. Draw AB, construct 60° at A, mark C 4.5 cm along the arm, then join BC.
Equilateral, isosceles and right-angled triangles
An equilateral triangle uses one radius for every arc. An isosceles triangle uses the same radius from both ends of the base. A right-angled triangle needs a 90° construction at one vertex.
Constructing Circles
Set the compasses to the radius with a ruler, place the point on the centre and turn the compasses through a full turn without changing the gap. If you are given the diameter, halve it first: a circle of diameter 9 cm has radius 4.5 cm.
Original Practice Questions With Worked Answers
- Which angle do you bisect to get 45°? 90°.
- Describe how to construct 30°. Construct 60° by stepping one arc of the same radius along the first arc, then bisect the 60°.
- Which two constructed angles do you bisect between to get 105°? 90° and 120°.
- How do you get 22.5°? Bisect 45° (90° bisected twice).
- Why do the stepped arcs give 60°? The centre and the two arc points form an equilateral triangle, whose angles are all 60°.
- A triangle has sides 8 cm, 8 cm and 5 cm. What type is it and how do you construct it? Isosceles. Draw the 5 cm base, then draw arcs of radius 8 cm from each end to meet at the apex.
Practise each construction on plain paper and check with a protractor. Grade 8 learners extend these skills in the Grade 8 geometrical constructions notes. Timed questions are in the Grade 7 CBC Challenge. These notes are free; full revision papers come with the KSh 20 Day Pass or the KSh 500 Monthly Pass. Free calculators and other revision aids are on the CBC Guides tools page.
Frequently Asked Questions
How do you construct a 90° angle with a ruler and compasses?
Construct 60° and 120° from the same point on a line, then bisect the angle between the two arms. The bisector makes 90° with the line.
What does it mean to bisect an angle?
To divide it into two equal angles using compass arcs, so each half is exactly half the original angle.
Can you use a protractor in a construction question?
Only to check. The question expects a ruler and a pair of compasses, and the construction arcs must be shown.
How do you construct 105°?
Construct 90° and 120° at the same point and bisect the angle between them, giving 90° + 15° = 105°.