Construct a 60° angle
An equilateral triangle gives a 60° angle.
Which triangle has 60° angles?
We need a 60° angle with only a ruler and a compass. There is a triangle whose three angles are all 60°.
Which triangle is it?
What this lesson covers
The idea
To construct a 60° angle at A on AX, draw an arc centred at A cutting AX at B, then an equal-radius arc from B meeting it at C; triangle ABC is equilateral, so ∠CAX = 60°.
Which triangle has 60° angles?
We need a 60° angle with only a ruler and a compass. There is a triangle whose three angles are all 60°.
Which triangle is it?
- The equilateral triangle
- A right-angled triangle
- Any isosceles triangle
Build an equilateral triangle
Draw an arc from A across the line at B. With the same opening, draw an arc from B. They meet at C. Then measure the angle at A.
Three equal sides, three 60° angles
AB and AC are radii of the first arc. BC is a radius of the second arc, with the same opening. So AB = AC = BC: triangle ABC is equilateral, and its angles are equal. They add up to 180°, so each is 180° ÷ 3 = 60°.
To construct a 60° angle at A on AX, draw an arc centred at A cutting AX at B, then an equal-radius arc from B meeting it at C. ∆ABC is equilateral, so ∠CAX = 60°.
- Side | Why it is the same length
- AB | radius of the arc from A
- AC | radius of the arc from A
- BC | radius of the arc from B, same opening
Notes
To construct a 60° angle at A on AX, draw an arc centred at A cutting AX at B, then an equal-radius arc from B meeting it at C. ∆ABC is equilateral, so ∠CAX = 60°.
Check yourself
Why is ∠CAX equal to 60°?
The same compass opening is used for the arc from A and the arc from B. They meet at C. The triangle ABC has a perimeter of 12 cm. How many cm is BC?
Answer: 4
All three sides equal the opening of the compass, so each is 12 ÷ 3 = 4 cm.
Bisect a 60° angle, and then bisect one of the halves. How many degrees is the smaller part?
Answer: 15
60° ÷ 2 = 30°, and 30° ÷ 2 = 15°. So a 15° angle is also possible with a ruler and a compass.
Rita opens the compass to 3 cm for the arc from A, but to 4 cm for the arc from B. They meet at C. Is ∠CAX equal to 60°?
- ∆ABC has three equal sides, so it is equilateral and each angle is 180° ÷ 3 = 60°. — correct. Yes! All three sides equal the opening of the compass.
- Because the compass is opened to 60 mm.. The size of the opening does not matter. Any opening gives 60°.
- Because C is the highest point of the arc.. The height of C does not decide the angle. The equal sides do.
- Yes. Any two arcs that meet give 60°.. Only when the opening is the same for both arcs, so that the triangle is equilateral.
- Yes, because the angle is made at A.. The angle at A is 60° only in an equilateral triangle.
- No. AB = AC = 3 cm but BC = 4 cm, so ABC is not equilateral. — correct. Yes! The same opening must be used for both arcs.