Question

figure not drawn to scale. what is the perimeter of the incomplete circle above with radius 27? (a) 54π (b) 32.4π (c) 48π (d) 48π + 54

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 $\theta = 40^{\circ}$, and since $1^{\circ}=\frac{\pi}{180}$ radians, then $\theta = 40\times\frac{\pi}{180}=\frac{2\pi}{9}$ radians. With $r = 27$, the arc - length $s$ is $s=r\theta=27\times\frac{2\pi}{9}=6\pi$.

Step2: Find the length of the two radii

The two radii that form the incomplete part of the circle have a combined length of $2r$. Since $r = 27$, the combined length of the two radii is $2\times27 = 54$.

Step3: Find the perimeter of the incomplete circle

The perimeter $P$ of the incomplete circle is the sum of the arc - length and the combined length of the two radii. The full - circle arc corresponding to the non - missing part has a central angle of $360 - 40=320^{\circ}$. In radians, $320^{\circ}=320\times\frac{\pi}{180}=\frac{16\pi}{9}$ radians. The length of the non - missing arc is $r\theta=27\times\frac{16\pi}{9}=48\pi$. Then $P = 48\pi+54$.

Answer:

D. $48\pi + 54$