Step by step
The question
Question: find the area A of the sector of a circle with radius 4 and angle π/3.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 4.
And the angle: θ = π/3.
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: 4² = 16, not 4.
Area always has a square in it.
2 Put the numbers in
Now put the numbers in: r = 4 and θ = π/3.
4 squared is 16.
3 Work it out
Tops times tops: 16 × π is 16π.
Bottoms: 2 × 3 is 6.
16 and 6 can both be divided by 2.
16 ÷ 2 is 8.
6 ÷ 2 is 3.
So A = 8π/3 square units.
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 8.38 square units, to the nearest hundredth.
4 Check
Let's check another way.
π/3 is a part of the full turn, 2π: 1/6 of it.
The whole circle is π r squared: 16π.
1/6 of 16π is 8π/3.
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.