1. **State the problem:** We have a square with area 100 cm² inscribed inside a circle. We need to find the area of the circle. 2. **Recall formulas:** - Area of square: $A_{square} = s^2$ where $s$ is the side length. - Diagonal of square: $d = s\sqrt{2}$. - The diagonal of the square is the diameter of the circle. - Area of circle: $A_{circle} = \pi r^2$ where $r$ is the radius. 3. **Find the side length of the square:** $$s = \sqrt{A_{square}} = \sqrt{100} = 10 \text{ cm}$$ 4. **Find the diagonal (diameter of circle):** $$d = s\sqrt{2} = 10\sqrt{2}$$ 5. **Find the radius of the circle:** $$r = \frac{d}{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$$ 6. **Calculate the area of the circle:** $$A_{circle} = \pi r^2 = \pi (5\sqrt{2})^2 = \pi \times 25 \times 2 = 50\pi$$ 7. **Answer:** The area of the circle is $50\pi$ cm², which corresponds to option A.