The area of a circle is calculated using the formula π r², where r is the radius. Given the radius r = 6 cm, the area of the circle is π (6)² = 36π cm².

A circle of 6 radius cm is inscribed in a square. Find the area of the shaded region.

Answer: The area of the shaded region is $(144 - 36\pi) \text{ cm}^2$ .

Step-by-step solution

Step 1: Determine the side length of the square

When a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square. Since the radius of the circle is given as $6 \text{ cm}$ , the diameter will be $2 \times 6 = 12 \text{ cm}$ . Therefore, the side length of the square is $12 \text{ cm}$ .

Step 2: Calculate the area of the square

The area of a square is calculated by squaring its side length. Since the side length of the square is $12 \text{ cm}$ , its area will be $12^2 = 144 \text{ cm}^2$ .

Step 3: Calculate the area of the circle

The area of a circle is calculated using the formula $\pi r^2$ , where $r$ is the radius. Given the radius $r = 6 \text{ cm}$ , the area of the circle is $\pi (6)^2 = 36\pi \text{ cm}^2$ .

Step 4: Calculate the area of the shaded region

The shaded region is the area of the square minus the area of the inscribed circle. So, we subtract the area of the circle ( $36\pi \text{ cm}^2$ ) from the area of the square ( $144 \text{ cm}^2$ ). This gives us $144 - 36\pi \text{ cm}^2$ .

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