Step by step
The question
Question: find the area A of the sector of a circle with radius 12 and angle 5π/6.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 12.
And the angle: θ = 5π/6.
1 Write the rule
The rule for the area of a sector: A = ½r²θ.
θ must be in radians.
Why?
A full turn is 2π.
Then A is a half times r squared times 2π: π r squared, the whole circle.
A smaller angle gives a part of it.
Careful!
r is squared: 12² = 144, not 12.
Area always has a square in it.
2 Put the numbers in
Now put the numbers in: r = 12 and θ = 5π/6.
12 squared is 144.
3 Work it out
Tops times tops: 144 × 5π is 720π.
Bottoms: 2 × 6 is 12.
720 and 12 can both be divided by 12.
720 ÷ 12 is 60.
12 ÷ 12 is 1.
So A = 60π square units.
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 188.50 square units, to the nearest hundredth.
4 Check
Let's check another way.
5π/6 is a part of the full turn, 2π: 5/12 of it.
The whole circle is π r squared: 144π.
5/12 of 144π is 60π.
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.