Step by step
The question
Question: a sector has radius 10 cm and angle 270°: 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 270° out of 360°: that is 3/4 of the circle.
part = 270°360° = 34
270 and 360 can both be divided by 90.
270 ÷ 90 = 3, 360 ÷ 90 = 4
2 Whole circle
The whole edge, the circumference, is 2 π r: 2 times π times 10 is 20π.
circumference = 2 × π × 10 = 20π
3 Take the part
Now 3/4 of 20π: 15π.
arc length = 34 × 20π = 15π
In π that is 15π.
As a decimal: about 47.12 cm, to 2 decimal places.
≈ 15π ≈ 47.12 cm
4 Check
Let's check: the whole is 4/3 times the part: 20π again.
So it is right!
43 × 15π = 20π
Let's look back.
First: what part of the circle.
Then: whole circle.
Then: take the part.
Last: check.
Answer
≈ 15π ≈ 47.12 cm