Step by step
The question
Question: find the arc length s of a circle with radius 6 and angle 2π/3.
The arc is the curved part of the edge.
We know the radius: r = 6.
And the angle: θ = 2π/3.
1 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!
The rule only works with the angle in radians.
Ours already is.
2 Put the numbers in
Now put the numbers in: r = 6 and θ = 2π/3.
3 Work it out
6 times 2π is 12π.
The bottom stays 3.
12 and 3 can both be divided by 3.
12 ÷ 3 is 4.
3 ÷ 3 is 1.
So s = 4π.
That is the exact answer.
With the calculator, π is about 3.14159.
So s is about 12.57, to the nearest hundredth.
4 Check
Let's check another way.
2π/3 is a part of the full turn, 2π: 1/3 of it.
The whole edge is 2π r: 12π.
1/3 of 12π is 4π.
The same answer!
So it is right.
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.