Question

in circle u, uv = 8 and the length of \\( \overarc{vw} \\) = 2\\( \pi \\). find the area shaded below. express your answer as a fraction times \\( \pi \\).

Explanation:

Step1: Find the circumference of the circle

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(r=UV = 8\), so \(C=2\pi\times8=16\pi\).

Step2: Find the fraction of the un - shaded arc

The length of arc \(\overset{\frown}{VW}=2\pi\). The fraction of the un - shaded arc is \(\frac{\text{Arc length}}{\text{Circumference}}=\frac{2\pi}{16\pi}=\frac{1}{8}\).

Step3: Find the fraction of the shaded area

The area of a circle is \(A=\pi r^{2}\). The fraction of the shaded area is \(1-\frac{1}{8}\) (since the area of a sector is proportional to the length of its arc). \(1-\frac{1}{8}=\frac{8 - 1}{8}=\frac{7}{8}\).

Step4: Calculate the area of the shaded region

The area of the whole circle \(A=\pi r^{2}=\pi\times8^{2}=64\pi\). The shaded area \(A_{\text{shaded}}=\frac{7}{8}\times64\pi\).

Answer:

\(56\pi\)