Question

a circle is placed in a square with a side length of 12 mm, as shown below. 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: Calculate the area of the square

The area formula for a square is \(A = s^2\), where \(s\) is the side - length. Given \(s = 12\) mm, then \(A_{square}=12\times12 = 144\) \(mm^{2}\).

Step2: Calculate the area of the circle

The area formula for a circle is \(A=\pi r^{2}\). Since the diameter of the circle is equal to the side - length of the square (\(d = 12\) mm), the radius \(r=\frac{d}{2}=\frac{12}{2}=6\) mm. Using \(\pi = 3.14\), then \(A_{circle}=3.14\times6^{2}=3.14\times36 = 113.04\) \(mm^{2}\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A = A_{square}-A_{circle}\). Substitute the values: \(A=144 - 113.04=30.96\) \(mm^{2}\).

Answer:

\(30.96\) \(mm^{2}\)