Question

How can you decompose the composite figure to determine its area?

as a pentagon and four semicircles
as two rectangles and four circles
as a square and four semicircles
as two triangles and four circles

Answer 1

You would decompose it as a square plus four semicircles.

Reason: the central polygon is a square and each side has a semicircle whose diameter is that side. So the composite area = area(square) + sum of the four semicircles. Since four semicircles = two full circles, with radius r = s/2 (s = side length of the square),

Area = s^2 + 2·π·r^2 = s^2 + 2π(s/2)^2 = s^2 + (π/2)s^2 = s^2(1 + π/2).