Final answer:
To find the area of the shaded region in a square with a quarter circle inscribed, calculate the area of the full circle using the radius (half the side of the square), divide it by four for the quarter circle, and then subtract this area from the area of the square. The result is 12.566 cm² to three significant figures.
Step-by-step explanation:
To find the area of the shaded region, we first need to understand the relationship between the square and the quarter circle that is inscribed within it. The side of the square is given as 4 cm, which will be the diameter of the circle that the quarter circle comes from. Therefore, the radius of the circle is half the side of the square, which is 2 cm.
To calculate the area of the full circle, we use the formula A = πr². Plugging in our values:
A = π(2 cm)² = 4π cm²
Since we have a quarter circle, the area of the quarter circle is one-fourth of the full circle:
Area of quarter circle = ¼ × 4π cm² = π cm²
The area of the full square is the side length squared:
Area of the square = (4 cm)² = 16 cm²
To obtain the area of the shaded region, we subtract the area of the quarter circle from the area of the square:
Area of shaded region = 16 cm² - π cm² = (16 - π) cm²
Using a calculator and keeping in mind significant figures, we get:
Area of shaded region = 12.566 cm² to three significant figures.