sectors

Figure 1 shows a sector AOB of a circle with centre O and radius r cm. The angle AOB is θ radians. The area of the sector AOB is 11 cm^2. Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of r.

What is the length of the arc AB in terms of $r$ and $\theta$ ?

so the perimeter is made up of 2 radii and the curved bit....

P = 2r + rθ

we know that P = 4x rθ

and that the area A = r² x θ

so r² x θ = 11

now you can find r and θ

I don't get how

we know that

2r + rθ = 4rθ

because the perimeter ( LHS ) is 4 times the arc length ( RHS )

by rearranging it you can find θ

and we know that

0.5 r²θ = 11

you can then put in the value of θ and find r

I get stuck after 2r = 3r(theta)

You can cancel out the $r$ since it's non-zero, and hence get what $\theta$ is.

Then sub it into the area equation and solve that for r.

we know that

2r + rθ = 4rθ

because the perimeter ( LHS ) is 4 times the arc length ( RHS )

by rearranging it you can find θ

and we know that

0.5 r²θ = 11

you can then put in the value of θ and find r