Question

Find the area and the perimeter of the shaded regions below. Give your answer as a completely simplified, exact value in terms of PI (no approximations). The figures are based on semicircles or quarter circles, and problems (b), (c), and (d) involve portions of a square. Calculate area and perimeter

Answer 1

Check the picture below.

`\stackrel{\textit{\Large Areas}}{\stackrel{triangle}{\cfrac{1}{2}(6)(6)}~~ + ~~\stackrel{semi-circle}{\cfrac{1}{2}\pi (3)^2}}\implies \boxed{18+4.5\pi} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{pythagorean~theorem}{CA^2 = AB^2 + BC^2\implies} CA=√(AB^2 + BC^2) \\\\\\ CA=√(6^2+6^2)\implies CA=√(6^2(1+1))\implies CA=6√(2) \\\\\\ \stackrel{\textit{\Large Perimeters}}{\stackrel{triangle}{(6+6√(2))}~~ + ~~\stackrel{semi-circle}{\cfrac{1}{2}2\pi (3)}}\implies \boxed{6+6√(2)+3\pi}`

notice that for the perimeter we didn't include the segment BC, because the perimeter of a figure is simply the outer borders.