1. **Problem statement:** A square is inscribed in a circle with area $\pi$ square units. Find the area of the square. 2. **Given:** Area of circle $= \pi$. 3. **Step 1: Find the radius of the circle.** The area of a circle is given by the formula: $$\text{Area} = \pi r^2$$ Given the area is $\pi$, we have: $$\pi r^2 = \pi$$ Divide both sides by $\pi$: $$\cancel{\pi} r^2 = \cancel{\pi}$$ $$r^2 = 1$$ Taking the positive root (radius is positive): $$r = 1$$ 4. **Step 2: Relate the radius to the square's side length.** The square is inscribed in the circle, so the circle passes through all four vertices of the square. The diagonal of the square equals the diameter of the circle. Diameter $= 2r = 2 \times 1 = 2$ 5. **Step 3: Find the side length of the square.** If $s$ is the side length of the square, then by the Pythagorean theorem: $$\text{diagonal} = s\sqrt{2}$$ Set diagonal equal to diameter: $$s\sqrt{2} = 2$$ Divide both sides by $\sqrt{2}$: $$s = \frac{2}{\sqrt{2}}$$ Rationalize the denominator: $$s = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$$ 6. **Step 4: Calculate the area of the square.** $$\text{Area of square} = s^2 = (\sqrt{2})^2 = 2$$ **Final answer:** The area of the square is $2$ square units. **Note:** The mistake in your approach was assuming the side length of the square is twice the radius. Actually, the diagonal of the square equals the diameter, not the side length. So the side length is $\sqrt{2}$, not $2$.