CONSTRUCTIONS (Using ruler and compasses only)
173. The 60-Degree Blueprint · Two equal arcs lock in a perfect angle
The angle ∠CBE is always exactly 60° — the equilateral triangle BDE locks it in.
A 60° angle can be constructed because an equilateral triangle has all angles equal to 60°. Draw an arc of radius r from vertex B — it cuts the base at D. Draw another arc of the same radius from D — it cuts the first arc at E. Since BD = BE = DE = r, triangle BDE is equilateral, so ∠CBE = 60° exactly.
What this lesson covers
Try to break it
Drag B anywhere on the canvas. The whole construction follows, but the angle at B always reads exactly 60°. BD = BE = DE (all equal to radius r) forms an equilateral triangle — so 60° is locked in permanently. Try to break it.
How you build it
Construct a 60° angle using compass and ruler.
- Place point B anywhere on the canvas — this is the vertex of the 60° angle you will construct.
- Place point C to the right of B. C marks the direction of the base arm — ray BC.
- Ray tool: click B, then click C. This draws ray BC — the base arm of the 60° angle.
- Arc tool: click centre B, then drag to set radius r. The arc sweeps across ray BC and will cross it at a point — you will mark that crossing as D next.
- Place point D exactly where the arc crosses ray BC. D is the anchor for the second arc — it must sit right on the crossing.
- Arc tool: click centre D. The radius r from step 4 is locked. The arc swings upward and crosses the B-arc at a point — mark that crossing as E next.
- Place point E exactly where the B-arc and D-arc cross each other. E is the key point — the ray through B and E gives the 60° angle.
- Ray tool: click B, then click E. This is the second arm of the angle. Since BD = BE = DE = r, triangle BDE is equilateral — so ∠CBE = 60°.
The proof, step by step
Prove that the constructed angle ∠CBE equals 60°.
- By construction, BD and BE are both radii of the arc drawn from B with radius r. So BD = BE = r.
- The arc from D was drawn with the same radius r, so DE = r. Therefore BD = BE = DE = r.
- Triangle BDE has all three sides equal (= r), so it is equilateral.
- All angles of an equilateral triangle are 60°. Hence ∠DBE = 60°, which is the same as ∠CBE = 60°.
Worked example
A student constructs a 60° angle using ruler and compass. She then bisects this angle. What is the measure of each of the two new angles formed?
Bisecting a 60° angle divides it into two equal parts: 60° ÷ 2 = 30°. Each new angle measures 30°. This is how compass-and-ruler construction of a 30° angle works — construct 60° first, then bisect.
- 20°
- 30° — correct
- 45°
- 60°