Step by step
The question
Question: a sector has radius 6 cm and angle 90°: 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 90° out of 360°: that is 1/4 of the circle.
part = 90°360° = 14
90 and 360 can both be divided by 90.
90 ÷ 90 = 1, 360 ÷ 90 = 4
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 1/4 of 12π: 3π.
arc length = 14 × 12π = 3π
Careful!
The perimeter is all the way round the slice: the arc and the two straight radii.
Careful: not perimeter = 3π ✗
4 Add the two radii
All the way round: the arc plus two radii, 6 and 6.
That is 3π plus 12, about 21.42 cm.
perimeter = 3π + 6 + 6 = 3π + 12 ≈ 21.42 cm
5 Check
Let's check: 4 slices like this make the whole circle.
4 times 3π is 12π, the whole edge.
So it is right!
4 slices: 4 × 3π = 12π
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 = 3π + 6 + 6 = 3π + 12 ≈ 21.42 cm