Step by step
The question
Question: find the arc length s of a circle with radius 12 m and angle 150°.
The arc is the curved part of the edge.
We know the radius: r = 12 m.
And the angle: θ = 150°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
150° is 150 times one degree.
So we multiply 150 by π/180.
Top times top: 150π.
The bottom stays 180.
150 and 180 can both be divided by 30.
It is the biggest number that goes into both.
150 ÷ 30 is 5.
180 ÷ 30 is 6.
So 150° = 5π/6 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.
12 times 150 gives 1800: much longer than the whole edge of the circle!
3 Put the numbers in
Now put the numbers in: r = 12 and θ = 5π/6.
4 Work it out
12 times 5π is 60π.
The bottom stays 6.
60 and 6 can both be divided by 6.
60 ÷ 6 is 10.
6 ÷ 6 is 1.
So s = 10π m.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 31.42 m, to the nearest hundredth.
5 Check
Let's check another way.
150° is a part of the full turn, 360°: 5/12 of it.
The whole edge is 2π r: 24π.
5/12 of 24π is 10π.
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.