Three angles, three triangles
Three angles give triangles that are easy to measure.
- A circle, radius 7.
- 60°: equilateral. Chord = radius.
- 90°: two legs at a right angle.
- 120°: two 30-60-90 halves, height 7/2.
The rule
Each triangle has two radii of 7 as sides. The angle between them is at the centre. The chord is the third side.
| Angle | The triangle | Chord | Area |
|---|---|---|---|
| 60° | Equilateral | 7 | 49√3/4 |
| 90° | Two legs at a right angle | 7√2 | 49/2 |
| 120° | Two 30-60-90 halves | 7√3 | 49√3/4 |
At 60°, the two radii are equal. So the other two angles are equal. They add to 180° − 60° = 120°, so each is 60°. All three sides equal 7.
Split the equilateral down the middle. Each half is a right triangle. Its hypotenuse, the longest side, is 7. Its short side is 7/2.
Height² = 7² − (7/2)² = 49 − 49/4 = 147/4. So the height is 7√3/2. Area = ½ × 7 × 7√3/2 = 49√3/4.
At 90°, the radii are the legs of a right triangle. The chord is opposite that corner, so it is the hypotenuse. Chord = √(49 + 49) = 7√2. Area = ½ × 7 × 7 = 49/2.
At 120°, the radii are equal. So the other two angles are equal. They add to 180° − 120° = 60°, so each is 30°. Draw a line from the centre to the chord's middle. It meets the chord at 90° and halves the 120° angle.
Each half has angles 30°, 60° and 90°, and hypotenuse 7. The short side is half the hypotenuse. So the height is 7/2.
The half-chord is √(49 − 49/4) = 7√3/2. So the whole chord is 7√3. Area = ½ × 7√3 × 7/2 = 49√3/4.
The 60° and 120° areas match. For one radius, the 60° and 120° triangles have equal area.
Read the chord squared. Tap the angle.
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