Final answer:
To find the area of the region lying outside the circle and inside the square, subtract the area of the circle from the area of the square. The area is approximately 6.72 cm².
Step-by-step explanation:
To find the area of the region lying outside the circle and inside the square, we need to subtract the area of the circle from the area of the square. Since the square's diagonal is 8 cm, we can use the Pythagorean theorem to find the length of a side of the square. Dividing the diagonal by √2, we get 8/√2 = 8√2/2 = 4√2 cm. The area of the square is then (4√2 cm)^2 = 32 cm².
The radius of the circle is half the length of a side of the square, which is 4√2 cm / 2 = 2√2 cm. The area of the circle is given by the formula A = πr², so the area of the circle is π(2√2 cm)² = 8π cm².
Finally, we can find the area of the region lying outside the circle and inside the square by subtracting the area of the circle from the area of the square: 32 cm² - 8π cm². Using an approximation of π as 3.14, we can calculate the area to be approximately 6.72 cm².