To find the area of the square inscribed in a circle, you need to use the relationship between the radius of the circle and the diagonal of the square.

Steps:

1. Radius of the circle (r) = 7 cm.
2. The diagonal of the square is equal to the diameter of the circle.
   • The diameter of the circle is: $\text{Diameter} = 2r = 2 \times 7 = 14 \, \text{cm}$.
3. Let the side length of the square be $s$. Using the Pythagorean theorem for the diagonal of the square, we know: $\text{Diagonal of the square} = s\sqrt{2}$ Since the diagonal of the square is also the diameter of the circle, we have: $s\sqrt{2} = 14$
4. Solve for $s$ (the side length of the square): $s = \frac{14}{\sqrt{2}} = 14 \times \frac{\sqrt{2}}{2} = 14 \times 0.7071 = 9.899 \, \text{cm}$
5. Now, to find the area of the square: $\text{Area of the square} = s^2 = (9.899)^2 = 98 \, \text{cm}^2$

Thus, the area of the square inscribed in the circle is approximately 98 cm².

Would you like more details or have any questions? Here are 5 related questions you might find interesting:

1. How would the side length change if the radius of the circle was doubled?
2. How can we derive the formula for the area of the square inscribed in a circle from first principles?
3. What happens to the area of the square if the circle's radius is halved?
4. How do the areas of the square and circle compare when the square is inscribed in the circle?
5. Can you calculate the area of the circle with the given radius?

Tip: The diagonal of an inscribed square always equals the diameter of the circle, which is key to finding the side length.