Question about perimeter of a sector of a circle

Question

I need to find the radius. The answer says "$3.79$ or $3.793$ to $3.974$" but I am not quite sure how to get there.

Why is this method wrong: $$3(65/360 × 2π × r)+ 6r = 20.5.$$ (Sorry for poor editing)

Answer 1

You're almost there, but take a look at the figure one more time. Why do you add $6r$? There are $3$ circle sectors, and $2$ straight lines (equal in length to the radius). The length of the perimeter should therefore be $$3(\frac{65}{360}\times2\pi r)+2r=20.5$$

Answer 2

Looking at the diagram, if the radius is $R$ and the perimeter is $P$, we get that the perimeter consists of $2$ radiuses (left and right), and $3$ parts of the perimeter.

$$2R + 3(2\pi R \times \frac{65^\circ}{360^\circ}) = P$$

I assume you can figure out the rest yourself.