Question

Area of the square which can be drawn in the circle of radius 4 cm will be

• A. 64 cm\(^2\)
• B. 32 cm\(^2\)
• C. 128 cm\(^2\)
• D. 256 cm\(^2\)

Hint:

Knowing the formula for the area of a square in terms of its diagonal (Area = \( \frac{1}{2} d^2 \)) is a very useful shortcut for problems involving inscribed squares in circles. It saves you the step of calculating the side length.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The largest square that can be drawn inside a circle is an inscribed square. The vertices of this square lie on the circumference of the circle. The diagonal of the inscribed square is equal to the diameter of the circle.
Step 2: Key Formula or Approach:
1. Diameter of a circle = 2 \( \times \) Radius.
2. The diagonal (\(d\)) of an inscribed square is equal to the diameter of the circle.
3. The area of a square can be calculated using its side (\(s\)) as Area = \(s^2\), or using its diagonal (\(d\)) as Area = \( \frac{1}{2} d^2 \).
Step 3: Detailed Explanation:
The radius of the circle is given as 4 cm.
First, calculate the diameter of the circle:
\[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 4 = 8 \text{ cm} \] The diagonal of the inscribed square is equal to the diameter of the circle.
\[ d = 8 \text{ cm} \] Now, calculate the area of the square using the diagonal formula:
\[ \text{Area} = \frac{1}{2} d^2 \] \[ \text{Area} = \frac{1}{2} (8)^2 \] \[ \text{Area} = \frac{1}{2} \times 64 \] \[ \text{Area} = 32 \text{ cm}^2 \] Step 4: Final Answer:
The area of the square is 32 cm\(^2\).