Step by step
The question
Question: find the area A of the sector of a circle with radius 5 and angle 7π/4.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 5.
And the angle: θ = 7π/4.
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: 5² = 25, not 5.
Area always has a square in it.
2 Put the numbers in
Now put the numbers in: r = 5 and θ = 7π/4.
5 squared is 25.
3 Work it out
Tops times tops: 25 × 7π is 175π.
Bottoms: 2 × 4 is 8.
So A = 175π/8 square units.
That is the exact answer.
With the calculator, π is about 3.14159.
So A is about 68.72 square units, to the nearest hundredth.
4 Check
Let's check another way.
7π/4 is a part of the full turn, 2π: 7/8 of it.
The whole circle is π r squared: 25π.
7/8 of 25π is 175π/8.
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.