A circle is inscribed in a square whose diagonal is 6√2 cm. Using properties of a square and its inscribed circle, what is the area of this circle (in square centimetres)?

Introduction / Context:
This question combines properties of squares with circles inscribed in them (incircles). Knowing the diagonal of the square allows you to find its side length, and hence the diameter and radius of the inscribed circle. Then you can compute the area of the circle using the standard formula πr².

Given Data / Assumptions:

• A square has diagonal length d = 6√2 cm.
• A circle is inscribed inside the square, touching all four sides.
• We are asked to find the area of the circle in square centimetres.
• The square is assumed to be perfect, with right angles and equal sides.

Concept / Approach:
For a square with side s and diagonal d: d = s√2 ⇒ s = d / √2 For an inscribed circle, the diameter equals the side of the square, so radius r = s / 2. The area of the circle is: Area = πr^2

Step-by-Step Solution:
Given d = 6√2. Compute side s = d / √2 = (6√2) / √2 = 6 cm. For the inscribed circle, diameter = s = 6 cm. So radius r = s / 2 = 6 / 2 = 3 cm. Now area of the circle = πr^2 = π * 3^2. That is area = 9 π square centimetres.

Verification / Alternative check:
We can double check the diagonal: if s = 6 cm, then diagonal should be s√2 = 6√2, which matches the given diagonal. This confirms that side and hence radius are correctly determined. The circle area π * 9 is therefore reliable.

Why Other Options Are Wrong:
6 π and 3 π correspond to radii √6 and √3, which do not match the geometry of an incircle in this square. 9√2 π would involve an extra √2 factor and would arise only if the radius were incorrectly taken as 3√2 instead of 3.

Common Pitfalls:
A typical mistake is to confuse the diagonal with the side, or to treat the diagonal as the diameter of the circle. Another is failing to divide the side by 2 to get the radius. Always remember: inscribed circle diameter equals the side of the square, not the diagonal.

Final Answer:
The area of the inscribed circle is 9 π square centimetres.