Step by step
The question
Question: find the arc length s of a circle with radius 3 and angle 90°.
The arc is the curved part of the edge.
We know the radius: r = 3.
And the angle: θ = 90°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
90° is 90 times one degree.
So we multiply 90 by π/180.
Top times top: 90π.
The bottom stays 180.
90 and 180 can both be divided by 90.
It is the biggest number that goes into both.
90 ÷ 90 is 1.
180 ÷ 90 is 2.
So 90° = π/2 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.
3 times 90 gives 270: much longer than the whole edge of the circle!
3 Put the numbers in
Now put the numbers in: r = 3 and θ = π/2.
4 Work it out
3 times π is 3π.
The bottom stays 2.
So s = 3π/2.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 4.71, to the nearest hundredth.
5 Check
Let's check another way.
90° is a part of the full turn, 360°: 1/4 of it.
The whole edge is 2π r: 6π.
1/4 of 6π is 3π/2.
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.