Question

a rectangle is placed around a semicircle as shown below. the width of the rectangle is 4 cm. find the area of the shaded region. use the value 3.14 for π, and do not round your answer. be sure to include the correct unit in your answer.

Explanation:

Step1: Find length of rectangle

The width of the rectangle is the radius of the semicircle, so radius \( r = 4 \) cm. The length of the rectangle is the diameter of the semicircle, so length \( l = 2r = 8 \) cm.

Step2: Calculate area of rectangle

Area of rectangle \( A_{rect} = length \times width = 8 \times 4 = 32 \) \( cm^2 \).

Step3: Calculate area of semicircle

Area of a full circle is \( \pi r^2 \), so area of semicircle \( A_{semicircle} = \frac{1}{2} \pi r^2 \). Substituting \( r = 4 \) and \( \pi = 3.14 \), we get \( A_{semicircle} = \frac{1}{2} \times 3.14 \times 4^2 = \frac{1}{2} \times 3.14 \times 16 = 25.12 \) \( cm^2 \).

Step4: Find area of shaded region

Shaded area = Area of rectangle - Area of semicircle = \( 32 - 25.12 = 6.88 \) \( cm^2 \).

Answer:

\( 6.88 \) square centimeters