Step by step
The question
Question: find the area A of the sector of a circle with radius 2 and angle 3 radians.
A sector is a slice of the circle, like a slice of pizza.
We know the radius: r = 2.
r = 2
And the angle: θ = 3 radians.
θ = 3
1 Write the rule
The rule for the area of a sector: A = ½r²θ.
θ must be in radians.
A = 12r2θ
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.
θ = 2π: 12 × r² × 2π = πr²
Careful!
r is squared: 2² = 4, not 2.
Area always has a square in it.
Careful: not A = 12 × 2 × θ ✗
2 Put the numbers in
Now put the numbers in: r = 2 and θ = 3.
A = 12 × 22 × 3
2 squared is 4.
A = 12 × 4 × 3
3 Work it out
Half of 4 is 2.
Times 3: 6.
So A = 6 square units.
A = 6
4 Check
Let's check: 2A/r squared gives the angle back.
2 × 64 = 3
We get 3: the angle.
So it is right!
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
Answer
A = 6