Step by step
The question
Question: find the arc length s of a circle with radius 15 cm and angle 40°.
The arc is the curved part of the edge.
We know the radius: r = 15 cm.
And the angle: θ = 40°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
40° is 40 times one degree.
So we multiply 40 by π/180.
Top times top: 40π.
The bottom stays 180.
40 and 180 can both be divided by 20.
It is the biggest number that goes into both.
40 ÷ 20 is 2.
180 ÷ 20 is 9.
So 40° = 2π/9 radians.
2 Write the rule
The rule for the arc length: s = rθ.
θ must be in radians.
Why?
A full turn is 2π.
Then s is 2π times r: the whole edge of the circle.
A smaller angle gives a part of it.
Careful!
θ must be in radians.
15 times 40 gives 600: much longer than the whole edge of the circle!
3 Put the numbers in
Now put the numbers in: r = 15 and θ = 2π/9.
4 Work it out
15 times 2π is 30π.
The bottom stays 9.
30 and 9 can both be divided by 3.
30 ÷ 3 is 10.
9 ÷ 3 is 3.
So s = 10π/3 cm.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 10.47 cm, to the nearest hundredth.
5 Check
Let's check another way.
40° is a part of the full turn, 360°: 1/9 of it.
The whole edge is 2π r: 30π.
1/9 of 30π is 10π/3.
The same answer!
So it is right.
Let's look back.
First: change to radians.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.