Constructions with a Ruler and Compasses
Ruler and compass constructions let you draw perpendicular bisectors, angle bisectors and accurate triangles without measuring, and they appear regularly on the Edexcel IGCSE Maths papers. On this page every construction is animated step by step, so you can watch each compass arc appear in order and then try the moves yourself with the on-screen compass, protractor and tracing paper. When you know the methods, scroll down to the auto-marked practice questions to test recognising constructions, equidistance facts and the angles they produce.
What are constructions?
A construction is an accurate drawing made with only two tools: a ruler (used as a straight edge) and a pair of compasses. No measuring of angles is allowed; the compass arcs do all the work. For centuries this was how precise technical drawing was done, and Edexcel IGCSE exam questions still ask you to construct bisectors, perpendiculars and triangles with these two tools.
The golden rules:
- Use a sharp pencil and a stiff pair of compasses that will not slip.
- Never rub out your construction arcs. The arcs are the evidence that you constructed rather than measured, and they carry method marks.
- Keep the compasses at the same radius throughout a step. Equal radii are what make each construction exact.
- Draw arcs generously long, so crossing arcs really do cross.
Step-by-step constructions with a ruler and compasses
Each panel below draws one construction a step at a time, exactly as you would on paper: compass arcs first, then the ruler line. Press Next to advance at your own pace, and read the final step to see why the construction works.
Perpendicular bisector of a line segment
- Draw the line segment AB with a ruler.
- Open the compasses to more than half of AB. With the point on A, draw arcs above and below the line.
- Keep the same radius. With the point on B, draw two more arcs, and mark both crossing points.
- Join the crossing points with a ruler: this line is the perpendicular bisector, cutting AB in half at right angles.
- Why it works: every point P on this line is the same distance from A as from B (the tick marks show PA = PB).
Angle bisector
- Start with the angle you want to cut in half.
- With the compass point on the vertex, draw one arc that crosses both arms. Mark the two crossing points.
- From each crossing point, keeping the same radius, draw a small arc in the middle of the angle. Mark where they cross.
- Join the vertex to that crossing point: this line is the angle bisector, splitting the angle into two equal halves.
- Why it works: every point Q on the bisector is the same perpendicular distance from the two arms (tick marks equal).
Perpendicular from a point to a line
- Start with a line and a point P that is not on the line.
- With the compass point on P, draw an arc that cuts the line twice. Mark the two cuts.
- From each cut, same radius, draw an arc on the far side of the line. Mark where these arcs cross.
- Join P to that crossing point: the join meets the line at right angles.
- Why it matters: this perpendicular is the shortest distance from P to the line, which is what exam questions usually want.
Perpendicular at a point on a line (90°, then 45°)
- Start with a line and a point P on it.
- With the compass point on P, draw arcs cutting the line on both sides of P, and mark the two cuts.
- Open the compasses wider. From each cut, draw an arc above the line, and mark where the arcs cross.
- Join P to the crossing point: a perfect 90° angle at P.
- Bisect one of the right angles (dashed line) and you have constructed 45°.
Triangle from three given sides (SSS)
- Make a rough sketch of the triangle and label the sides, so you know what you are aiming for. Here the sides are 9 cm, 7 cm and 5 cm.
- Draw the base accurately with a ruler: 9 cm from A to B.
- Open the compasses to 7 cm and draw an arc from A.
- Open the compasses to 5 cm and draw an arc from B, crossing the first arc. The crossing point is C.
- Join A to C and B to C. The arcs guarantee AC = 7 cm and BC = 5 cm exactly, with no angle measuring at all.
A 60° angle, bisected to 30°
- Draw a base line and mark the vertex A.
- Draw an arc from A crossing the base line. Mark the crossing point.
- Keep the same radius. Draw an arc from the crossing point, cutting the first arc.
- Join A to the new crossing: a perfect 60° angle, because the two arcs build an equilateral triangle.
- Bisect the 60° angle (dashed line) to construct 30°. The same trick turns 90° into 45°.
To construct a triangle from two sides and the included angle (SAS) or two angles and the included side (ASA), you are allowed a protractor: draw the given side, measure the given angle(s) with the protractor, then finish with the ruler. The practice rooms below ask you to do exactly that in your exercise book, then measure your result.
Try the compass yourself
This practice area works with a mouse or one finger on a phone; everything you need to know is written under the panel.
🧭 The compass: sweep your own arcs
Your task: construct the perpendicular bisector of AB. AB is 7.5 cm, so set the opening to more than 3.75 cm. Drag the compass by its point to sit exactly on A (it clicks into place), tap Pencil: down, then drag the pencil tip to sweep an arc above and below the line. Tap the pencil button again to lift it, move the point to B, and sweep again with the same opening. The crossings give the bisector.
The animated panel above shows this same construction step by step if the interactive tool does not load.
Worked examples
💡 Can these sides make a triangle?
Priya tries to construct a triangle with sides 10 cm, 4 cm and 5 cm, using the 10 cm side as the base. Explain why the construction is impossible.
What's happening?
The two arcs have radii 4 cm and 5 cm, drawn from the ends of a 10 cm base. Even pointing straight at each other they only reach 9 cm, so the arcs never cross and no triangle exists. The two shorter sides together must be longer than the base.
💡 Angles from repeated bisection
An angle of 84° is bisected, and one of the halves is bisected again. What angles are produced?
What's happening?
Each bisection halves the angle exactly: first two angles of 42°, then the second bisector splits one 42° into two of 21°. Bisection is exact because of the equal compass radii, not approximate like measuring.
💡 Using the equidistance fact
P is a point on the perpendicular bisector of AB, and PA = 6.5 cm. Write down the length PB, giving a reason.
PB = 6.5 cm
Reason: every point on the perpendicular bisector of AB is equidistant from A and B.
What's happening?
No calculation is needed. Being on the perpendicular bisector means being the same distance from both ends, so PB must equal PA. This one fact answers a whole family of exam questions.
💡 Choosing the right construction
A lamp must be fixed the same distance from two straight fences that meet at a corner. Which construction shows where the lamp can go?
The angle bisector of the corner angle.
What's happening?
Same distance from two lines (the fences) means the angle bisector. If instead it had to be the same distance from two points (say, two posts), you would construct the perpendicular bisector of the segment joining them. Deciding "points or lines?" chooses the construction.
🔑 Key Points
- Leave every arc visible: construction marks carry method marks.
- Same radius within a step; for a perpendicular bisector the opening must be more than half the segment.
- Perpendicular bisector = all points equidistant from the two ends; angle bisector = all points equidistant from the two arms.
- The perpendicular from a point to a line is the shortest distance to the line.
- Constructible angles without a protractor: 60°, 30°, 15°, 90°, 45° (construct, then bisect).
- SSS triangles need no protractor; SAS and ASA use one for the given angles.
⚠️ Common Pitfalls
- Rubbing out arcs to make the page "neat": the marks are the method.
- Letting the compasses slip to a new radius halfway through a bisector.
- Arcs drawn too short to cross: sweep generously past where you expect the crossing.
- Measuring the midpoint or the half-angle by eye or with a protractor when the question says construct: that scores no method marks.
- Forgetting the triangle test: the two shorter sides together must beat the longest side, or the arcs never meet.