The share again
A sprinkler swings through part of a circular lawn.
- A circle of radius r.
- A wedge of θ degrees.
- Arc = θ/360 of the way round.
- Sector = θ/360 of the area inside.
The rule
An arc is a part of the circle's rim, its curved edge. A sector is the wedge between two radii and an arc. Here θ (theta) is the wedge's angle at the centre. A full turn is 360°. So a wedge of θ degrees is θ/360 of a turn. The arc is that same share of the rim. The sector is that same share of the area inside.
Arc length = θ/360 × 2πr. Sector area = θ/360 × πr².
Here r is the radius. It runs from the centre to the rim.
| Angle | Share | Arc | Sector |
|---|---|---|---|
| 90° | 1/4 | 11 | 38.5 |
| 60° | 1/6 | 22/3 | 77/3 |
For radius 7, 2πr = 2 × 22/7 × 7 = 44. Also πr² = 22/7 × 49 = 154.
At 90°: arc = 44 ÷ 4 = 11. Sector = 154 ÷ 4 = 38.5.
Take the sprinkler again. It swings through 60°, radius 7. Its share is 60/360, which is 1/6 of the turn. Its arc is 44 ÷ 6, which is 22/3. Its sector is 154 ÷ 6, which is 77/3.
A sector smaller than half the circle is a minor sector. The rest of the circle is the major sector. Major sector = πr² − minor sector.
Use one π in a whole sum. Here that π is 22/7. Some questions say π = 3.14. Then use 3.14 from the first line. Do not change it halfway through.
Slide the angle. Watch the arc and sector change together.
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