Question

the diameter of a circle is 6 inches. what is the area of a sector bounded by a 120° arc?
d=6 in
120°
give the exact answer in simplest form.
square inches

Explanation:

Step1: Find the radius

The radius \(r\) of a circle is half of the diameter. Given \(d = 6\) inches, so \(r=\frac{d}{2}=\frac{6}{2}=3\) inches.

Step2: Use the sector - area formula

The formula for the area of a sector of a circle is \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(\theta\) is the central angle of the sector and \(r\) is the radius of the circle. Here, \(\theta = 120^{\circ}\) and \(r = 3\) inches.
Substitute the values into the formula: \(A=\frac{120^{\circ}}{360^{\circ}}\times\pi\times(3)^{2}\).
Simplify \(\frac{120^{\circ}}{360^{\circ}}=\frac{1}{3}\), and \((3)^{2}=9\).
Then \(A=\frac{1}{3}\times\pi\times9\).
Multiply \(\frac{1}{3}\times9 = 3\).

Answer:

\(3\pi\) square inches