Step by step
The question
Question: find the area A of the sector of a circle with radius 10 and angle 72°.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 10.
And the angle: θ = 72°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
72° is 72 times one degree.
So we multiply 72 by π/180.
Top times top: 72π.
The bottom stays 180.
72 and 180 can both be divided by 36.
It is the biggest number that goes into both.
72 ÷ 36 is 2.
180 ÷ 36 is 5.
So 72° = 2π/5 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: 10² = 100, not 10.
Area always has a square in it.
3 Put the numbers in
Now put the numbers in: r = 10 and θ = 2π/5.
10 squared is 100.
4 Work it out
Tops times tops: 100 × 2π is 200π.
Bottoms: 2 × 5 is 10.
200 and 10 can both be divided by 10.
200 ÷ 10 is 20.
10 ÷ 10 is 1.
So A = 20π square units.
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 62.83 square units, to the nearest hundredth.
5 Check
Let's check another way.
72° is a part of the full turn, 360°: 1/5 of it.
The whole circle is π r squared: 100π.
1/5 of 100π is 20π.
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.