Questions: 1. The circle below has an area of 16π square units and a circumference of 8π units. The measures of the central angle θ, sector area AS', and arc length s are labeled. a. What is the ratio of the central angle measure, θ, to the total interior angle measure?

1. The circle below has an area of 16π square units and a circumference of 8π units. The measures of the central angle θ, sector area AS', and arc length s are labeled. a. What is the ratio of the central angle measure, θ, to the total interior angle measure?

Transcript text: 1. The circle below has an area of $16 \pi$ square units and a circumference of $8 \pi$ units. The measures of the central angle $\theta$, sector area $A_{S^{\prime}}$, and arc length $s$ are labeled. a. What is the ratio of the central angle measure, $\theta$, to the total interior angle measure?

Solution

Solution Steps

Step 1: Find the radius

The area of a circle is given by $A = \pi r^2$ and the circumference is given by $C = 2\pi r$, where $r$ is the radius. We are given that the area is $16\pi$, so $16\pi = \pi r^2$ $r^2 = 16$ $r = 4$

The circumference is given as $8\pi$, so $8\pi = 2\pi r$ $r = \frac{8\pi}{2\pi}$ $r = 4$

Step 2: Ratio of central angle to total interior angle

The total interior angle measure of a circle is $2\pi$ radians or $360^\circ$. The central angle is given as $\theta = \frac{\pi}{3}$ radians. The ratio of the central angle measure to the total interior angle measure is: $\frac{\theta}{2\pi} = \frac{\frac{\pi}{3}}{2\pi} = \frac{\pi}{3} \cdot \frac{1}{2\pi} = \frac{1}{6}$