Step by step
The question
Question: find the arc length s of a circle with radius 4 and angle 330°.
The arc is the curved part of the edge.
We know the radius: r = 4.
And the angle: θ = 330°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
330° is 330 times one degree.
So we multiply 330 by π/180.
Top times top: 330π.
The bottom stays 180.
330 and 180 can both be divided by 30.
It is the biggest number that goes into both.
330 ÷ 30 is 11.
180 ÷ 30 is 6.
So 330° = 11π/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.
4 times 330 gives 1320: much longer than the whole edge of the circle!
3 Put the numbers in
Now put the numbers in: r = 4 and θ = 11π/6.
4 Work it out
4 times 11π is 44π.
The bottom stays 6.
44 and 6 can both be divided by 2.
44 ÷ 2 is 22.
6 ÷ 2 is 3.
So s = 22π/3.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 23.04, to the nearest hundredth.
5 Check
Let's check another way.
330° is a part of the full turn, 360°: 11/12 of it.
The whole edge is 2π r: 8π.
11/12 of 8π is 22π/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.