The share again
Show on a big screenA 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.
Loading lab…
Try the lab to finish this lesson.