A square is inscribed in a circle. If each side of the square is 14 cm, what is the area (in square centimetres) of the circle that just encloses this square?

Introduction / Context:
The given problem connects the geometry of a square and a circle. A square is inscribed in a circle, which means that all four vertices of the square lie on the circle. The question asks for the area of the circle when the side length of the square is known. This tests understanding of the relationship between the diagonal of a square and the diameter of its circumcircle, as well as the basic circle area formula.

Given Data / Assumptions:
- The figure is a square inscribed in a circle.- Side of the square = 14 cm.- All vertices of the square lie on the circle.- The diagonal of the square is equal to the diameter of the circle.- We use the area formula for a circle: Area = π * r^2.

Concept / Approach:
The key concept is that for a square inscribed in a circle, the diagonal of the square becomes the diameter of the circle. Once the diameter is known, the radius is half of the diameter. Then, we use the standard area formula of a circle. We also use the Pythagorean relationship for the diagonal of a square: diagonal = side * √2. Substituting the given side into this relation gives us the diameter, from which we obtain the radius.

Step-by-Step Solution:
Step 1: Let the side of the square be s = 14 cm.Step 2: The diagonal of the square is given by diagonal = s * √2.Step 3: Substitute s = 14 cm to get diagonal = 14 * √2 cm.Step 4: Since the square is inscribed in the circle, the diagonal of the square equals the diameter of the circle. So diameter of the circle = 14 * √2 cm.Step 5: Radius r of the circle = (diameter) / 2 = (14 * √2) / 2 = 7 * √2 cm.Step 6: Area of the circle = π * r^2 = π * (7 * √2)^2.Step 7: Compute (7 * √2)^2 = 7^2 * (√2)^2 = 49 * 2 = 98.Step 8: Therefore, the area of the circle = 98π square centimetres.

Verification / Alternative check:
An alternative way is to note that the area of the circle will always be π times the square of half the diagonal of the square. Since the diagonal is 14√2, half of it is 7√2, which matches our radius. Squaring 7√2 again gives 98, so the area is 98π. The same result confirms that no computational mistake has been made.

Why Other Options Are Wrong:
- 49π: This would correspond to a radius of 7 cm instead of 7√2 cm, which ignores the √2 factor from the diagonal.- 77π: This does not match any correct radius obtained from the diagonal of the square and is not consistent with the geometry.- 121π: This would require r^2 = 121, that is r = 11, which is not related to the given side length 14 cm in this configuration.- 64π: This would correspond to r^2 = 64 and r = 8, again not matching the derived radius 7√2.

Common Pitfalls:
Students often mistakenly assume that the side of the square is the diameter of the circle instead of the diagonal. Another frequent error is forgetting to divide the diagonal by 2 to get the radius before applying the area formula. Some may also incorrectly square 7√2 and get 49√2 instead of 98, which arises from not squaring both the number and the square root properly. Careful handling of the relationship between side, diagonal, diameter, and radius avoids these issues.

Final Answer:
The area of the circle is 98π square centimetres.