Problem

The diagram shows a symmetrical plate $ABCDEF$. The line $ABCD$ is straight and the length of $BC$ is $2$ cm. Each of the two sectors $ABF$ and $DCE$ is of radius $r$ cm and each of the angles $ABF$ and $DCE$ is equal to $\frac{1}{3}\pi$ radians.

(a) It is given that $r = 0.4$ cm.

(i) Show that the length $EF = 2.4$ cm.

(ii) Find the area of the plate. Give your answer correct to $3$ significant figures.

(b) It is given instead that the perimeter of the plate is $6$ cm.

Find the value of $r$. Give your answer correct to $3$ significant figures.