Tools and rules of construction
In a construction we use only two tools:
- a compass, to draw circles and arcs, and to copy a length;
- a straightedge (ruler), to draw straight lines. We may use the scale to draw a given length, but not to measure a result.
Use a sharp pencil and leave all arcs visible; they show your method. A protractor is for checking, not for constructing.
Perpendicular bisector and perpendicular from a point
Perpendicular bisector of AB (a line that cuts AB in half at 90°):
- Open the compass to more than half of AB.
- From A, draw arcs above and below AB.
- Keep the same width; from B, draw arcs that cross the first ones at P and Q.
- Join PQ. It meets AB at the midpoint M.
Why it works: P and Q are the same distance from A and from B. Every point on PQ is equidistant from A and B. This line is a locus: the set of all points that follow one rule.
Perpendicular from a point P to a line: from P draw an arc cutting the line at two points; then construct the perpendicular bisector of those two points. It passes through P.
Angle bisector and special angles (60°, 30°, 90°, 45°)
Angle bisector of angle O: draw an arc from O cutting both arms at X and Y. From X and Y draw equal arcs meeting at Z. Join OZ. Every point on OZ is the same distance from both arms.
60°: draw a ray from O. With any radius draw an arc from O cutting the ray at A. With the same radius, draw an arc from A cutting the first arc at B. Angle AOB = 60°, because triangle OAB has three equal sides (equilateral).
- 30° = bisect 60°.
- 120° = step the radius twice.
- 90° = perpendicular, or bisect between 60° and 120°.
- 45° = bisect 90°.
Constructing triangles and the triangle inequality
Three sides (SSS): draw the longest side as the base. From one end, draw an arc with the second length; from the other end, an arc with the third length. Where they cross is the third vertex.
Two sides and the included angle (SAS): draw one side, construct the angle at one end, mark the second side along it, and join.
Two angles and a side (ASA): draw the side, construct both angles at its ends; the rays meet at the third vertex.
A triangle is possible only if the sum of any two sides is greater than the third (triangle inequality). If 2 + 3 ≤ 6, the arcs never meet.
Regular polygons and constructions in design
Regular hexagon: draw a circle of radius r. Starting at any point, step the same radius round the circle; it fits exactly 6 times. Join the points. Each side equals r, and each angle at the centre is 360° ÷ 6 = 60°.
- Join every other point: equilateral triangle.
- Two perpendicular diameters give a square; bisect the right angles to get a regular octagon.
Designers combine these to make tiles, rosettes, logos and technical drawings.
Key formulas and definitions
- Perpendicular bisector: all points equidistant from A and B
- Angle bisector: all points equidistant from both arms
- Radius steps round a circle 6 times → 60° at the centre
- 30° = ½ × 60°; 45° = ½ × 90°; 90° = 60° + 30°
- Triangle inequality: a + b > c for every pair of sides
- Central angle of a regular n-gon = 360° ÷ n
- Interior angle of a regular n-gon = (n − 2) × 180° ÷ n
Worked examples
1. Construct the perpendicular bisector of AB = 7 cm and find AM.
1) Draw AB = 7 cm. 2) Open the compass to about 4.5 cm (more than 3.5). 3) Arcs from A above and below. 4) Same width, arcs from B crossing at P and Q. 5) Join PQ, meeting AB at M. AM = MB = 3.5 cm and PQ ⟂ AB.
2. Construct an angle of 75°.
75° = 60° + 15°, or halfway between 60° and 90°. 1) Construct 60° and 90° at O on the same ray. 2) Bisect the angle between the 60° and 90° lines. The bisector makes 75° with the ray.
3. Can you construct a triangle with sides 3 cm, 4 cm and 8 cm?
No. 3 + 4 = 7, which is less than 8. The arcs of 3 cm and 4 cm from the ends of the 8 cm base never meet.
4. Construct triangle PQR with PQ = 6 cm, ∠P = 60°, PR = 4 cm (SAS).
1) Draw PQ = 6 cm. 2) Construct 60° at P. 3) Set the compass to 4 cm and mark R on that ray. 4) Join QR.
5. Construct a regular hexagon of side 3 cm. What are its interior angles?
1) Draw a circle of radius 3 cm. 2) Step 3 cm round the circle six times. 3) Join the points. Interior angle = (6 − 2) × 180° ÷ 6 = 120°.
6. Construct triangle ABC with BC = 5 cm, ∠B = 45°, ∠C = 60° (ASA). What is ∠A?
1) Draw BC = 5 cm. 2) At B construct 45° (bisect 90°). 3) At C construct 60°. 4) The rays meet at A. ∠A = 180° − 45° − 60° = 75°.
Common mistakes
- Changing the compass width between the arcs from A and from B. Both must be equal.
- Opening the compass to less than half of AB, so the arcs do not meet.
- Rubbing out the construction arcs. Examiners look for them.
- Using a protractor to draw the angle in a construction question. Construct it with a compass, then check.