Step by step
The question
Question: a sector has radius 8 and arc length 4π: find the angle of the sector.
The arc is the curved part of the edge of the slice.
1 Whole circle
The whole edge is 2 π r: 16π.
whole edge = 2 × π × 8 = 16π
2 What part of the circle
The sector is 4π out of 16π.
The π's cancel.
part = 4π16π = 14
4 goes into 16: 4 × 4 = 16.
So 4 goes into both.
We divide both by 4.
So it is 1/4 of the circle.
4 × 4 = 16
4 ÷ 4 = 1
16 ÷ 4 = 4
3 Find the angle
The slice is the same part of the turn as of the edge.
The whole turn is 360°.
So the angle is 1/4 of 360°.
angle = 14 × 360°
Cut the whole into 4 equal parts.
One part is 360 divided by 4: 90.
The π stays.
360 ÷ 4 = 90
So the angle is 90°.
angle = 90°
Careful!
The slice is the same part of the edge and of the angle.
Careful: the part of the circle = the part of 360°
4 Check
Let's check: 90 out of 360 is 1/4 too.
So it is right!
90°360° = 14
Let's look back.
First: whole circle.
Then: what part of the circle.
Then: find the angle.
Last: check.
Answer
angle = 90°