Question

For the figures below, assume they are made of semicircles, quarter circles and squares. For each shape, find the area and perimeter. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

Area of the shaded region $=36(\pi -2)$ square cm

Perimeter of the shaded region $=6 (\pi + 2\sqrt 2)$ cm

Solution:

Radius of the quarter of circle = 12 cm

Area of the shaded region = Area of quarter of circle – Area of the triangle

                                             $=\frac{1}{4} \pi r^2 - \frac{1}{2} bh$

                                             $=\frac{1}{4} \pi \times 12^2 - \frac{1}{2} \times 12 \times 12$

                                             $=36\pi -72$

                                             $=36(\pi -2)$ square cm.

Area of the shaded region $=36(\pi -2)$ square cm

Using Pythagoras theorem,

$AC^2=AB^2+BC^2$

$AC^2=12^2+12^2$

$AC^2=288$

Taking square root on both sides of the equation, we get

$AC= 12\sqrt 2$ cm

Perimeter of the quadrant of a circle = $\frac{1}{4} \times 2\pi r$

                                                             $=\frac{1}{4} \times 2 \times \pi \times 12$

                                                             $=6 \pi$ cm

Perimeter of the shaded region = $6 \pi + 12\sqrt 2$ cm

                                                    $=6 (\pi + 2\sqrt 2)$ cm

Hence area of the shaded region $=36(\pi -2)$ square cm

Perimeter of the shaded region $=6 (\pi + 2\sqrt 2)$ cm