A flat plate S, which is part of a child’s toy, is shown in the diagram above. The points \(A,B\) and \(C\) are the vertices of an equilateral triangle and the distance between \(A\) and \(B\) is \(2a\). The circular arc \(AB\) has centre \(C\) and radius \(2a\). The circular arc \(BC\) and \(CA\) have centres at \(A\) and \(B\) respectively and radii \(2a\).

(a) Find, in terms of \(\pi\) and \(a\), the perimeter of \(S\).

(b) Prove, that the area of the plate \(S\) is

  \(2a^2(\pi-\sqrt{3})\).

Were you able to prove this?