Question

a rectangle is placed around a semicircle as shown below. the width of the rectangle is 2 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 the length and width of the rectangle

The width of the rectangle is 2 cm (which is the radius of the semicircle, \( r = 2 \) cm). The length of the rectangle is twice the radius, so \( l = 2r = 2\times2 = 4 \) cm.

Step2: Calculate the area of the rectangle

The formula for the area of a rectangle is \( A_{rectangle} = l\times w \). Substituting the values, we get \( A_{rectangle} = 4\times2 = 8 \) \( cm^2 \).

Step3: Calculate the area of the semicircle

The formula for the area of a full circle is \( A_{circle} = \pi r^2 \), so the area of a semicircle is \( A_{semicircle} = \frac{1}{2}\pi r^2 \). Substituting \( r = 2 \) cm and \( \pi = 3.14 \), we get \( A_{semicircle} = \frac{1}{2}\times3.14\times2^2 = \frac{1}{2}\times3.14\times4 = 6.28 \) \( cm^2 \).

Step4: Calculate the area of the shaded region

The area of the shaded region is the area of the rectangle minus the area of the semicircle. So, \( A_{shaded} = A_{rectangle} - A_{semicircle} = 8 - 6.28 = 1.72 \) \( cm^2 \).

Answer:

\( 1.72 \) square centimeters (or \( 1.72 \, cm^2 \))