Step by step
The question
Question: a sector has radius 9 cm and angle 40°: 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 40° out of 360°: that is 1/9 of the circle.
part = 40°360° = 19
40 and 360 can both be divided by 40.
40 ÷ 40 = 1, 360 ÷ 40 = 9
2 Whole circle
The whole edge, the circumference, is 2 π r: 2 times π times 9 is 18π.
circumference = 2 × π × 9 = 18π
3 Take the part
Now 1/9 of 18π: 2π.
arc length = 19 × 18π = 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: 9 slices like this make the whole circle.
9 times 2π is 18π, the whole edge.
So it is right!
9 slices: 9 × 2π = 18π
Let's look back.
First: what part of the circle.
Then: whole circle.
Then: take the part.
Last: check.
Answer
≈ 2π ≈ 6.28 cm