Problem

The diagram shows the design for a company’s new logo. The sector of the circle, centre $O$, has radius $r$ cm. The acute angle $AOC=\frac{1}{3}\pi$ radians. The quadrilateral $OABC$ is a rhombus.

(a) Find an expression for the perimeter of the design. Give your answer in terms of $\pi$ and $r$.

(b) It is now given that the perimeter of the design is $200$ cm.

Find the area of the design. Give your answer to $3$ significant figures.