Step by step
The question
Question: find the area A of the sector of a circle with radius 8 cm and angle 45°.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 8 cm.
And the angle: θ = 45°.
1 Change to radians
The rule needs the angle in radians.
So we change it first.
45° is 45 times one degree.
So we multiply 45 by π/180.
Top times top: 45π.
The bottom stays 180.
45 and 180 can both be divided by 45.
It is the biggest number that goes into both.
45 ÷ 45 is 1.
180 ÷ 45 is 4.
So 45° = π/4 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: 8² = 64, not 8.
Area always has a square in it.
3 Put the numbers in
Now put the numbers in: r = 8 and θ = π/4.
8 squared is 64.
4 Work it out
Tops times tops: 64 × π is 64π.
Bottoms: 2 × 4 is 8.
64 and 8 can both be divided by 8.
64 ÷ 8 is 8.
8 ÷ 8 is 1.
So A = 8π cm².
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 25.13 cm², to the nearest hundredth.
5 Check
Let's check another way.
45° is a part of the full turn, 360°: 1/8 of it.
The whole circle is π r squared: 64π.
1/8 of 64π is 8π.
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.