Question

In the figure given above, \( ABCD \) is a square, and a circle is inscribed in it. All sides of the square touch the circle. If \( AB = 14 \, \text{cm} \), find the area of the shaded region. 

Hint:

When a circle is inscribed in a square, the diameter of the circle equals the side length of the square. Use this relationship to compute the radius of the circle.

Answer 1

Step 1: The area of the square is given by: \[ \text{Area of square} = (\text{side})^2. \] Substituting \( \text{side} = 14 \, \text{cm} \): \[ \text{Area of square} = 14^2 = 196 \, \text{cm}^2. \] Step 2: The area of the circle is given by: \[ \text{Area of circle} = \pi r^2. \] Here, the diameter of the circle is equal to the side of the square, so the radius \( r = \frac{14}{2} = 7 \, \text{cm} \). Substituting \( r = 7 \): \[ \text{Area of circle} = \frac{22}{7} \times 7 \times 7 = 154 \, \text{cm}^2. \] Step 3: The area of the shaded region is the difference between the area of the square and the area of the circle: \[ \text{Area of shaded region} = \text{Area of square} - \text{Area of circle}. \] Substituting the values: \[ \text{Area of shaded region} = 196 - 154 = 42 \, \text{cm}^2. \]