1. Calculate the circumference of a circle given its area 130.1 cm². The area of a circle is given by the formula $$A = \pi r^2$$ where $r$ is the radius. Rearranging to find $r$: $$r = \sqrt{\frac{A}{\pi}}$$ Substitute $A = 130.1$: $$r = \sqrt{\frac{130.1}{\pi}}$$ Calculate $r$: $$r \approx \sqrt{41.43} \approx 6.44 \text{ cm}$$ Circumference $C$ is given by $$C = 2 \pi r$$ Calculate $C$: $$C = 2 \pi \times 6.44 \approx 40.47 \text{ cm}$$ 2. A circle is inscribed in a square, touching all four sides. Given the circle's area is 131.5 cm², find the side length of the square. Since the circle touches all four sides, the diameter of the circle equals the side length $s$ of the square. Find radius $r$ from area: $$r = \sqrt{\frac{131.5}{\pi}} \approx \sqrt{41.87} \approx 6.47 \text{ cm}$$ Diameter $d = 2r = 2 \times 6.47 = 12.94$ cm. Therefore, side length of square $s = 12.94$ cm. 3. A circle is inscribed in a square with area 139.7 cm². Find the area of the square. Find radius $r$: $$r = \sqrt{\frac{139.7}{\pi}} \approx \sqrt{44.47} \approx 6.67 \text{ cm}$$ Diameter $d = 2r = 13.34$ cm. Side length of square $s = d = 13.34$ cm. Area of square: $$s^2 = 13.34^2 = 177.9 \text{ cm}^2$$ 4. Two circles inside a rectangle touch each other and three sides of the rectangle. Each circle has area 6.7 cm². Find the area of the rectangle. Find radius $r$ of each circle: $$r = \sqrt{\frac{6.7}{\pi}} \approx \sqrt{2.13} \approx 1.46 \text{ cm}$$ Since circles touch each other and three sides, the rectangle's length is twice the diameter, and width equals diameter. Diameter $d = 2r = 2.92$ cm. Length $L = 2d = 5.84$ cm, Width $W = d = 2.92$ cm. Area of rectangle: $$A = L \times W = 5.84 \times 2.92 = 17.06 \text{ cm}^2$$ 5. Large circle area = 13.2 cm², small circle area = 7.1 cm². Find how many times larger the radius of the large circle is compared to the small circle. Radius large: $$r_L = \sqrt{\frac{13.2}{\pi}} \approx \sqrt{4.20} \approx 2.05 \text{ cm}$$ Radius small: $$r_S = \sqrt{\frac{7.1}{\pi}} \approx \sqrt{2.26} \approx 1.50 \text{ cm}$$ Ratio: $$\frac{r_L}{r_S} = \frac{2.05}{1.50} \approx 1.37$$ The radius of the large circle is approximately 1.37 times larger than the radius of the small circle.