Step by step
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