Constructions
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Construction Tools and Conventions
- A construction uses only a pair of compasses (for arcs and circles of a fixed radius) and a straightedge (for drawing straight lines only, not for measuring) - never a protractor.
- Keep the compass width exactly fixed while drawing each pair of arcs.
- Always leave construction arcs and lines visible - they show your method.
Constructing a 60° Angle
Since every angle in an equilateral triangle is 60°, constructing an equilateral triangle constructs a 60° angle.
Constructing a 60° angle using two arcs of equal radius
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Method:
- Step 1: Draw a straight line and mark a point A on it.
- Step 2: Open the compasses to any convenient radius. With the point on A, draw an arc crossing the line at B.
- Step 3: Keeping the same radius, put the compass point on B and draw an arc crossing the first arc at C.
- Step 4: Join A to C. Angle BAC = 60°.
Constructing a 30° Angle
A 30° angle is constructed by bisecting (splitting exactly in half) a 60° angle.
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Method:
- Step 1: Construct a 60° angle, BAC, as above.
- Step 2: With the compass point on A, draw an arc crossing both arms of the angle.
- Step 3: From each of those two crossing points, draw arcs of equal radius that intersect at a new point D.
- Step 4: Join A to D. Angle BAD = 30°.
Constructing a 45° Angle
A 45° angle is constructed by first making a right angle (90°), then bisecting it.
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Method:
- Step 1 (construct 90°): On a line through point A, draw an arc centred at A crossing the line on both sides. From each crossing point, draw arcs of equal (larger) radius that intersect above the line - join A to that intersection for a right angle.
- Step 2 (bisect): Use the same angle-bisecting method as for 30° - draw an arc across both arms of the right angle, then two intersecting arcs from those points, then join to the new intersection.
- Answer: The resulting angle is 45°.
Constructing an Equilateral Triangle
An equilateral triangle has 3 equal sides and 3 equal (60°) angles - the same arc method used for a 60° angle completes the whole triangle.
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Method:
- Step 1: Draw a line segment AB with the required side length.
- Step 2: With the compass point on A and radius AB, draw an arc above the line.
- Step 3: With the compass point on B and the same radius, draw another arc crossing the first at C.
- Step 4: Join A to C and B to C. Triangle ABC is equilateral.
Constructing a Square Inscribed in a Circle
A different method is needed for a square than for a hexagon: instead of stepping the radius around the circle, a square comes from constructing TWO perpendicular diameters.
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Method:
- Step 1: Draw a circle and mark a diameter across it (a straight line through the centre, touching the circle at both ends).
- Step 2: Construct the perpendicular bisector of this diameter, extending it in both directions until it also touches the circle - this gives a second diameter, at right angles to the first.
- Step 3: The two diameters now cross the circle at four points.
- Step 4: Join these four points in order. The resulting quadrilateral is a square.
Constructing a Regular Hexagon
A regular hexagon's side length always equals the radius of the circle it's inscribed in, which makes it straightforward to construct.
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Method:
- Step 1: Draw a circle with the required radius and mark its centre, O.
- Step 2: Choose any point on the circle as the first vertex, P1.
- Step 3: Keeping the compass set to the circle's radius, mark the next point P2 on the circle from P1.
- Step 4: Repeat around the circle to mark P3, P4, P5, and P6, each exactly one radius from the last.
- Step 5: Join each consecutive pair of points to complete the regular hexagon.
Real-World Applications
Precise constructions matter in many practical fields:
- Architecture and technical drawing: accurate angles and shapes without digital tools.
- Carpentry: marking precise joint angles.
- Surveying: establishing perpendiculars and fixed angles on-site.
- Art and design: creating exact geometric patterns.
- Tiling and mosaics: regular hexagon and triangle patterns.
Exam Tips
Common Mistakes
MistakeChanging the compass width partway through a construction
Fixkeep the compass width exactly fixed for each pair of arcs - changing it changes the result
MistakeUsing a protractor instead of compass-and-straightedge methods for a "construct" question
Fixconstruction questions require compass-and-straightedge methods only
MistakeErasing the construction arcs after finishing
Fixalways leave your construction arcs and lines visible - they show your method
MistakeConfusing bisecting an angle with bisecting a line segment
Fixcheck carefully whether you need to split an angle in half, or find the midpoint of a line
MistakeUsing an incorrect compass radius for a regular hexagon
Fixa regular hexagon's side length always equals the radius of its surrounding circle
MistakeTrying to construct an inscribed square by stepping the radius around the circle, like a hexagon
Fixa square uses TWO perpendicular diameters instead - a completely different method from the hexagon's
For Exams
- Leave your arcs visible - don't rub them out.
- Keep the same compass width throughout each individual arc-pair step.
- Work carefully with a sharp pencil for accurate constructions.
- Learn the key methods: 60° (equilateral triangle), 30°/45° (angle bisection), regular hexagon (radius = side length), square (perpendicular diameters).
- Practise both bisecting angles and bisecting lines - they use similar but different arc methods.
Interactive revision notes, videos and practice questions load below.