Question

The area of the circle that can be inscribed in a square of 6 cm is

• A. \(36\pi\text{ cm}^2\)
• B. \(15\pi\text{ cm}^2\)
• C. \(12\pi\text{ cm}^2\)
• D. \(9\pi\text{ cm}^2\)

Hint:

Remember the direct relation: Area of an inscribed circle in a square of side \(a\) is \(\pi \left(\frac{a}{2}\right)^2 = \frac{\pi a^2}{4}\).
With \(a = 6\), we get \(\frac{36\pi}{4} = 9\pi\text{ cm}^2\) instantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is from the topic of Mensuration, dealing with 2D shapes (circles and squares).
We need to calculate the area of a circle that is inscribed (fits perfectly inside) a square of a given side length.

Step 2: Key Formula or Approach:
When a circle is inscribed in a square of side length \(a\):
1. The diameter of the circle (\(d\)) is equal to the side length of the square:
\[ d = a \]
2. The radius of the circle (\(r\)) is half of the diameter:
\[ r = \frac{a}{2} \]
3. The area of the circle is given by:
\[ A = \pi r^2 \]

Step 3: Detailed Explanation:
Given side length of the square:
\[ a = 6\text{ cm} \]
Since the circle is inscribed inside this square, the maximum width (diameter) of the circle is equal to the side of the square:
\[ \text{Diameter } (d) = 6\text{ cm} \]
Now, calculate the radius (\(r\)) of the circle:
\[ r = \frac{d}{2} = \frac{6}{2} = 3\text{ cm} \]
Using the formula for the area of a circle:
\[ \text{Area} = \pi r^2 \]
Substitute the value of \(r\):
\[ \text{Area} = \pi (3)^2 \]
\[ \text{Area} = 9\pi\text{ cm}^2 \]

Step 4: Final Answer:
The area of the inscribed circle is \(9\pi\text{ cm}^2\).