Question

Find the area of the shaded region in figure, if ABCD is a square of side 14 cm, APD and BPC are semicircles.(use \(\pi = \frac{22}{7}\))

• A. 10.5 cm²
• B. 21 cm²
• C. 42 cm²
• D. 154 cm²

Answer 1

The Correct Option is C

To solve the problem, we are given that \(ABCD\) is a square of side \(14\, \text{cm}\), and arcs \(APD\) and \(BPC\) are semicircles. We need to find the area of the shaded region.

1. Calculate Area of the Square:
\[ \text{Area of square} = \text{side}^2 = 14 \times 14 = 196 \, \text{cm}^2 \]

2. Calculate the Area of the Two Semicircles:
Each semicircle has diameter \(14 \, \text{cm}\), so radius \(r = \frac{14}{2} = 7 \, \text{cm}\).

\[ \text{Area of one semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times \frac{22}{7} \times 7^2 = \frac{1}{2} \times \frac{22}{7} \times 49 = \frac{1078}{14} = 77 \, \text{cm}^2 \]
So, total area of two semicircles: \[ 2 \times 77 = 154 \, \text{cm}^2 \]

3. Subtract Semicircle Areas from Square Area:
\[ \text{Shaded area} = \text{Area of square} - \text{Area of semicircles} = 196 - 154 = 42 \, \text{cm}^2 \]

Final Answer:
The area of the shaded region is \({42 \, \text{cm}^2} \).