Question

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown. The area of the shaded region is \underline{\hspace{1cm} cm$^2$.} \includegraphics[width=0.5\linewidth]{image2.png}

• A. 8
• B. 4
• C. 12
• D. 10

Hint:

When arcs lie entirely below a semicircle of the same radius inside a square, the white region often reduces to the area of that semicircle. Then the shaded area is simply (square area) $-$ (semicircle area).

Answer 1

The Correct Option is D

Step 1: Identify the white (unshaded) portion.
The white portion is exactly the top semicircle drawn inside the square. Its diameter equals the side of the square ($4$ cm), hence radius $r=2$ cm.

Step 2: Area of the white semicircle.
\[ A_{\text{white}}=\frac{1}{2}\pi r^2=\frac{1}{2}\pi(2)^2=2\pi~\text{cm}^2. \]

Step 3: Area of the square.
\[ A_{\text{square}}=4\times 4=16~\text{cm}^2. \]

Step 4: Area of the shaded region.
Shaded area $=$ (area of square) $-$ (area of white semicircle): \[ A_{\text{shaded}}=16-2\pi\approx 16-6.283=9.717\ \text{cm}^2\ \approx 10\ \text{cm}^2. \] \[ \boxed{A_{\text{shaded}}=16-2\pi\ \text{cm}^2\ \approx 10\ \text{cm}^2} \]