Step by step
The question
Question: find the area A of the sector of a circle with radius 6 m and angle 120°.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 6 m.
And the angle: θ = 120°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
120° is 120 times one degree.
So we multiply 120 by π/180.
Top times top: 120π.
The bottom stays 180.
120 and 180 can both be divided by 60.
It is the biggest number that goes into both.
120 ÷ 60 is 2.
180 ÷ 60 is 3.
So 120° = 2π/3 radians.
2 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: 6² = 36, not 6.
Area always has a square in it.
3 Put the numbers in
Now put the numbers in: r = 6 and θ = 2π/3.
6 squared is 36.
4 Work it out
Tops times tops: 36 × 2π is 72π.
Bottoms: 2 × 3 is 6.
72 and 6 can both be divided by 6.
72 ÷ 6 is 12.
6 ÷ 6 is 1.
So A = 12π m².
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 37.70 m², to the nearest hundredth.
5 Check
Let's check another way.
120° is a part of the full turn, 360°: 1/3 of it.
The whole circle is π r squared: 36π.
1/3 of 36π is 12π.
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.