Question

The following shapes are based only on squares, semicircles, and quarter circles. Find the perimeter and area of the shaded part. Give your answer as a completely simplified exact value in terms of pi. (no approximations) FIRST PERSON TO ANSWER GETS 100 POINTS

Answer 1

Perimeter = 8π cm

Area = (32π - 64) cm²

Explanation:

ABCD is a square.

Therefore:

• AB = BC = CD = DA = 8 cm
• Area = 8 × 8 = 64 cm²

Perimeter

The perimeter of the shaded part is the sum of two congruent quarter circles, i.e. half the length of the circumference.

The arcs AC are quarter circles of a circle with radius 8 cm.

Therefore the perimeter of the shaded area is:

`\implies \sf Perimeter=Half\;circumference=(1)/(2) \cdot 2\pi r=\pi r=8\pi\;cm`

Area

To find the area of the shaded area, subtract the two unshaded areas from the area of the square.

As the two unshaded areas are congruent we only need to find the area of one unshaded area.

The area of one unshaded area is the area of the square minus the area of the sector ADC (which is a quarter of a circle with radius 8 cm).

`\begin{aligned}\implies \textsf{Area of one unshaded area}&=\textsf{Area of square}-\textsf{Area of $(1)/(4)$ circle}\\&=64-(1)/(4) \pi r^2\\&=64-(1)/(4) \pi \cdot 8^2\\&=64-(1)/(4) \pi \cdot 64\\&=(64-16 \pi)\; \sf cm^2\end{aligned}`

Now we have found the area of one unshaded area, to calculate the area of the shaded area simply subtract two unshaded areas from the area of the square:

`\begin{aligned}\implies \textsf{Shaded area}&=64-2(64-16\pi)\\&=64-128+32\pi\\&=(32\pi-64)\; \sf cm^2 \end{aligned}`