Worked examples
The question
Question: a sector has radius 6 cm and angle 60°: find the arc length.
The arc is the curved part of the edge of the slice.
1 What part of the circle
A whole turn is 360°.
The slice is 60° out of 360°: that is 1/6 of the circle.
part = 60°360° = 16
60 and 360 can both be divided by 60.
60 ÷ 60 = 1, 360 ÷ 60 = 6
2 Whole circle
The whole edge, the circumference, is 2 π r: 2 times π times 6 is 12π.
circumference = 2 × π × 6 = 12π
3 Take the part
Now 1/6 of 12π: 2π.
arc length = 16 × 12π = 2π
In π that is 2π.
As a decimal: about 6.28 cm, to 2 decimal places.
≈ 2π ≈ 6.28 cm
4 Check
Let's check: 6 slices like this make the whole circle.
6 times 2π is 12π, the whole edge.
So it is right!
6 slices: 6 × 2π = 12π
Let's look back.
First: what part of the circle.
Then: whole circle.
Then: take the part.
Last: check.
▶ Watch it solved
The question
Question: a sector has radius 5 cm and angle 72°: find the perimeter of the sector.
The arc is the curved part of the edge of the slice.
1 What part of the circle
A whole turn is 360°.
The slice is 72° out of 360°: that is 1/5 of the circle.
part = 72°360° = 15
72 and 360 can both be divided by 72.
72 ÷ 72 = 1, 360 ÷ 72 = 5
2 Whole circle
The whole edge, the circumference, is 2 π r: 2 times π times 5 is 10π.
circumference = 2 × π × 5 = 10π
3 Take the part
Now 1/5 of 10π: 2π.
arc length = 15 × 10π = 2π
Careful!
The perimeter is all the way round the slice: the arc and the two straight radii.
Careful: not perimeter = 2π ✗
4 Add the two radii
All the way round: the arc plus two radii, 5 and 5.
That is 2π plus 10, about 16.28 cm.
perimeter = 2π + 5 + 5 = 2π + 10 ≈ 16.28 cm
5 Check
Let's check: 5 slices like this make the whole circle.
5 times 2π is 10π, the whole edge.
So it is right!
5 slices: 5 × 2π = 10π
Let's look back.
First: what part of the circle.
Then: whole circle.
Then: take the part.
Then: add the two radii.
Last: check.
Answer perimeter = 2π + 5 + 5 = 2π + 10 ≈ 16.28 cm
▶ Watch it solved
The question
Question: a sector has radius 6 and area 12π: find the angle of the sector.
A sector is a slice of the circle, like a slice of pizza.
1 Whole circle
The whole circle has area π r squared: 36π.
whole area = π × 6² = 36π
2 What part of the circle
The sector is 12π out of 36π.
The π's cancel: 1/3 of the circle.
part = 12π36π = 13
3 Find the angle
The whole turn is 360°.
1/3 of 360° is 120°.
angle = 13 × 360° = 120°
Careful!
The slice is the same part of the area and of the angle.
Careful: the part of the circle = the part of 360°
4 Check
Let's check: 120 out of 360 is 1/3 too.
So it is right!
120°360° = 13
Let's look back.
First: whole circle.
Then: what part of the circle.
Then: find the angle.
Last: check.
Answer Careful: the part of the circle = the part of 360°
▶ Watch it solved
