Question

the diameter of a circle is 16 centimeters. what is the angle measure of an arc bounding a sector with area 6π square centimeters?
k=6π sq. cm

d=16 cm

give the exact answer in simplest form.

Explanation:

Step1: Calculate the radius of the circle

The formula for the radius \(r\) of a circle given the diameter \(d\) is \(r=\frac{d}{2}\). Given \(d = 16\) cm, then \(r=\frac{16}{2}=8\) cm.

Step2: Use the sector - area formula

The area formula of a sector is \(K=\frac{\theta}{360}\times\pi r^{2}\), where \(K\) is the area of the sector, \(\theta\) is the central angle (in degrees), and \(r\) is the radius of the circle.
We know \(K = 6\pi\) and \(r = 8\). Substitute these values into the formula:
\(6\pi=\frac{\theta}{360}\times\pi\times8^{2}\)
First, cancel out \(\pi\) on both sides of the equation:
\(6=\frac{\theta}{360}\times64\)
Then, solve for \(\theta\):
\(\theta=\frac{6\times360}{64}\)
Simplify the fraction:
\(\theta=\frac{2160}{64}=\frac{135}{4} = 33.75\)

Answer:

\(33.75\)