Question

figure not drawn to scale. what is the perimeter of the incomplete circle above with radius 9? (a) 16π + 18 (b) 16π (c) 14π + 18 (d) 14π

Explanation:

Step1: Find the arc - length

The formula for the length of an arc of a circle is $s = r\theta$, where $r$ is the radius and $\theta$ is the central - angle in radians. First, convert the angle from degrees to radians. Given $\theta = 80^{\circ}$, then $\theta=\frac{80\pi}{180}=\frac{4\pi}{9}$ radians. The full - circle central angle is $2\pi$ radians. The angle of the arc of the incomplete circle is $\theta_{arc}=2\pi-\frac{4\pi}{9}=\frac{18\pi - 4\pi}{9}=\frac{14\pi}{9}$ radians. With $r = 9$, the arc - length $s=r\theta_{arc}=9\times\frac{14\pi}{9}=14\pi$.

Step2: Add the lengths of the two radii

The two radii that form the open part of the incomplete circle have a combined length of $2r$. Since $r = 9$, the combined length of the two radii is $2\times9 = 18$.

Step3: Calculate the perimeter

The perimeter $P$ of the incomplete circle is the sum of the arc - length and the combined length of the two radii, so $P=14\pi+18$.

Answer:

C. $14\pi + 18$