Ruler-and-Compass Constructions

Perfect geometry from two humble tools

Long before laser levels and digital protractors, builders and stonemasons laid out perfect right angles and split angles dead in half using nothing but a length of string and a compass. That same craft — exact geometry from the simplest tools — is what a construction is.

Hand a mathematician a pair of compasses and an unmarked straight edge — no ruler markings, no protractor, no measuring at all — and they can still build a perfect right-angle, split any angle exactly in two, and drop a 60° angle onto the page with total precision. This is Euclidean construction: an art more than two thousand years old, in which the geometry itself does the measuring and the answer comes out exact rather than “near enough by eye”.

A construction is an accurate drawing made with only those two tools: the straight edge draws straight lines, and the compasses draw circles and arcs of a fixed radius. The magic is that equal radii create equal distances you can rely on — and from that single idea everything else follows.

The perpendicular bisector of a segment

Given a segment AB, the perpendicular bisector is the line that cuts it exactly in half and meets it at a right angle. To construct it:

Open the compasses wider than half of AB.
With that radius, draw arcs from A (above and below the line).
Keeping the same radius, draw arcs from B; they cross the first pair at two points.
Draw the straight line through those two crossing points.

That line is the perpendicular bisector. It works because the two crossing points are each the same distance from A as from B — and in fact every point on the line is equidistant from A and B. That is not a coincidence: the construction is a hands-on proof of the equidistant locus you meet in loci and regions.

Worked example 1 — bisect a segment

Say it in words, then step through it in the figure below:

Set the compasses wider than half of AB (any such width works).
Compass point on A: sweep a faint arc above the line and one below.
Without changing the width, move the compass point to B and sweep two more arcs. They cut the first pair at two points.
Lay the straight edge across those two crossings and rule the line.

Worked example 2 — bisect an angle

The same “equal arcs” trick splits an angle in half. Given an angle at a vertex V:

Put the compass point on V and draw an arc that crosses both arms of the angle.
From each of those two crossing points, draw equal arcs in the middle of the angle so they meet.
Rule the line from V through that meeting point: it splits the angle into two equal halves.

Every point on that bisector is equidistant from the two arms — the equidistant-from-two-lines locus, proved with a compass. A right angle bisected gives two 45° angles; a 60° angle bisected gives two 30° angles, and so on.

Worked example 3 — a 60° angle and an equilateral triangle

Compasses give you a perfect 60° angle for free, because an equilateral triangle has three 60° angles:

Draw a base line and mark a start point O on it.
Compass point on O: draw an arc that crosses the base at a point P.
Keep the same radius, move the compass point to P, and draw a second arc that crosses the first at Q.
Rule OQ. The angle QOP is exactly 60°, and OPQ is an equilateral triangle (all three radii are equal, so all three sides are equal).

The perpendicular from a point to a line

One more standard construction: the shortest path from a point P straight down onto a line.

Compass point on P: draw an arc that crosses the line at two points.
Those two points are now equidistant from P — so the perpendicular bisector of the segment joining them passes through P.
Construct that perpendicular bisector (equal arcs from each of the two points); it is the perpendicular from P to the line.

Each construction turns equal compass radii into a guarantee:

Perpendicular bisector — equal arcs from each end of AB meet at points equidistant from A and B; the line through them bisects AB at a right angle.
Angle bisector — equal arcs from the vertex (and from where they cross the arms) split the angle into two equal parts.
60° angle / equilateral triangle — three equal radii force three equal sides, so every angle is 60°.
SSS triangle — arc the two other side lengths from the ends of the base; their crossing is the third vertex.

Never change the compass width between the paired arcs. The whole method rests on the two arcs having the same radius — that is what makes the crossing points truly equidistant. Nudge the compasses wider (or narrower) between the arc from A and the arc from B and the symmetry breaks: your “bisector” will be slightly off and meet the line at not-quite-90°.
Do NOT rub out your construction arcs. The faint arcs are the evidence that you used a valid construction rather than a ruler or a lucky guess. In an exam the marks are awarded for those arcs — a clean final line with no arcs showing usually scores zero, even if it is perfectly correct.
“Wider than half” matters for the bisector. If the compass width is less than half of AB the two arcs never meet, and you get no crossing points at all.

Because compass-and-straightedge construction is so old — it comes straight from Euclid’s Elements, written around 300 BC — mathematicians spent literally two millennia stuck on three famous puzzles: squaring the circle (build a square with the same area as a given circle), trisecting an angle (split any angle into three equal parts), and doubling the cube. They were finally shown to be impossible with these tools — not merely hard, but provably out of reach — in the 1800s, using modern algebra. A 2000-year-old geometry question answered by brand-new algebra is one of the great crossovers in all of mathematics.