The twin segments
Two angles give the very same triangle area.
- 60° triangle: base 7, height 7√3/2.
- 120° triangle: base 7√3, height 7/2.
- Both cover 49√3/4. The twins.
The rule
Take a circle of radius 7. Cut it into six equal sectors. A full turn is 360°. So each has a 60° angle at the centre. One chord cuts a segment off one sector.
Sector (60°) = 154/6, which is 77/3. Triangle = 49√3/4.
Segment (60°) = 77/3 − 49√3/4 square units.
Both segments here are minor. So each is its sector minus its triangle.
√3 is not a fraction, and its decimal never ends. Keep it in the answer. A decimal in its place is not exact. When the question gives √3 = 1.73, use that value at the end.
The 120° sector is 120/360, which is 1/3, of 154. So it is 154/3. Its triangle is also 49√3/4 — the very same as the 60° triangle.
Segment (120°) = 154/3 − 49√3/4.
A segment's edge is its arc plus its chord. At 60° the arc is 44/6 = 22/3 and the chord is 7. The edge reads 22/3 + 7. At 120° the arc is 44/3 and the chord is 7√3. The edge reads 44/3 + 7√3.
Tap the area of each segment.
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