Question

Four circles are described at four corners of a square ABCD so that each touches two of the others. If the side of the square is 4 cm, find the area of the shaded region.

OPTION 1: 8π - 8
OPTION 2: 16 - 4π
OPTION 3: 16π - 16
OPTION 4: 8 - 2π

Answer 1

Option 2). To find the area of the shaded region, subtract the combined area of the four circles from the area of the square. The correct answer is Option 2: 16 - 4π.

Step-by-step explanation:

We must deduct the total area of the four circles from the area of the square in order to determine the area of the shaded area. Given that the square's side length is 4 cm, its area is 4 cm * 4 cm, or 16 cm². Since each circle is described at one of the square's corners, its radius is equal to half of the square's side length, or 2 cm. The formula A = π * r², where r is the radius, yields the area of each circle. The total area of the four circles is therefore equal to 4 * π * (2 cm)², or 16π cm². This can be subtracted from the square's area to get 16 cm² - 16π cm², or 16 - 4π cm². Therefore, the correct answer is Option 2: 16 - 4π.