Constructing Angles of 60° and 30°

1. The 60° angle hides in an equilateral triangle

An equilateral triangle has three equal sides and three equal angles. The three angles of every triangle add up to 180°, so each angle of an equilateral triangle is 180° ÷ 3 = 60°. This tells us how to construct a 60° angle: if we can draw an equilateral triangle, we automatically have 60° at each corner.

And we can draw an equilateral triangle using only compasses. Open the compasses to any radius. If you draw one arc from O and another arc from A with the same radius, the point B where they cross is exactly the same distance from O as from A, and OA is also that distance. So OA = OB = AB, and triangle OAB is equilateral.

2. Constructing an angle of 60°

Here is the standard method. A fine pencil and a firm compass hinge make this construction very accurate.

1. 1Draw a straight line and mark the point O where the angle will be.
2. 2With the compass point on O, draw a large arc that cuts the line at A.
3. 3Keep the same opening. Put the compass point on A and draw a second arc that cuts the first arc at B.
4. 4Use the ruler to join O and B. Angle AOB = 60°.

▶ Notice that the same opening is used from O and from A. That makes triangle OAB equilateral, so angle AOB = 60°.

You can also construct 60° at a point on a given line segment. If the segment is XY and you need 60° at X, treat X as the point O: draw an arc from X cutting XY at A, then the same radius from A, and so on. The 60° angle opens from the segment.

3. Constructing an angle of 30°

Thirty degrees is half of 60°. So to get 30°, construct 60° first and then bisect it. The bisecting steps are exactly the ones you learned in the lesson on copying and bisecting angles.

1. 1Construct angle AOB = 60° as above.
2. 2Draw an arc from A and an arc from B, both with the same opening, to cross at P.
3. 3Join O to P. Angle AOP = angle POB = 30°.

▶ Notice that the 30° angle is made by bisecting the 60° angle: OP is the angle bisector.

Notice that 30° can also be made by bisecting a 120° angle twice, but the route through 60° is quicker. In the next lesson you will bisect 30° to get 15° and then add 60° to get 75°.

4. Using 60° and 30° angles

These angles are important because they appear in many designs. The honeycomb is made of regular hexagons, whose corners are 120° (two 60° angles). Triangular roofs, truss frames and kente patterns use 60° triangles. A set square in your geometry box has a 30°-60°-90° shape.

• Six 60° angles fit exactly around a point (6 × 60° = 360°).
• Three 60° angles fit on a straight line (3 × 60° = 180°).
• Three 30° angles make a right angle (3 × 30° = 90°).
• A line from a corner of an equilateral triangle to the middle of the opposite side bisects the 60° angle, so it makes 30° on each side.

Any angle drawn by construction can be checked by measurement, and any angle given by measurement can be tested by calculation. Learn to move between the two: this is how engineers check that parts fit together.

Worked examples

Try each one yourself first, then open the solution and compare every step.