A circle is inscribed in a square whose diagonal measures 6√2 cm. What is the area of this inscribed circle (in square centimetres)?

Introduction / Context:
This problem combines properties of squares and circles. A circle inscribed in a square touches all four sides, so its diameter is equal to the side length of the square. The information is given in terms of the square diagonal instead of its side, requiring you to use the Pythagoras relation between diagonal and side before moving on to the circle. This two step structure is very typical in geometry based aptitude questions.

Given Data / Assumptions:

• A square has a diagonal of length 6√2 cm.
• A circle is inscribed in this square, touching all four sides.
• The diameter of the circle equals the side length of the square.
• We are asked to find the area of the circle in square centimetres.
• We use the standard formulas for squares and circles in Euclidean geometry.

Concept / Approach:
If the side of the square is s, then its diagonal d is given by d = s√2. We are given d and need to find s, so s = d / √2. Once we know s, we know the diameter of the circle because the circle just fits inside the square. Thus the radius r is s / 2. The area of the circle is then A = π * r^2. We will carefully follow these steps: find s, then find r, then find A.

Step-by-Step Solution:
Given diagonal d = 6√2 cm.For a square, d = s√2, so side s = d / √2 = (6√2) / √2 = 6 cm.The inscribed circle has diameter equal to the side of the square, so diameter = 6 cm, and radius r = 6 / 2 = 3 cm.Area of the circle A = π * r^2 = π * 3^2 = 9π sq cm.Thus the required area of the inscribed circle is 9π square centimetres.

Verification / Alternative check:
To double check, visualise the square of side 6 cm. The circle fits exactly inside, touching the midpoint of each side. The radius is clearly half the side length, which is 3 cm. Using any common approximation for π, such as 3.14 or 22/7, gives an area around 28.3 sq cm, but since the question keeps π symbolic, the exact expression 9π is correct. There are no other hidden steps, so the answer is reliable.

Why Other Options Are Wrong:
Option 6π would correspond to a radius √6, which is not consistent with the geometry here. Option 3π is far too small and would mean a radius of √3. Option 9√2π introduces an unnecessary √2 factor and does not arise from any correct combination of diagonal and radius. Option 12π is larger than 9π and does not match the radius 3 cm. Only 9π correctly reflects the square side and circle radius relation.

Common Pitfalls:
Students often forget that the diagonal of a square is side times √2, not side times 2. Another mistake is to treat the diagonal directly as the diameter of the circle, which would give a radius of 3√2 instead of 3. Some may also mix up radius and diameter, using d^2 instead of r^2 in the area formula. Keeping track of each geometric step prevents these errors.

Final Answer:
The area of the inscribed circle is 9π sq cm.