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?
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Solution

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Solution Steps

Step 1: Find the radius

The area of a circle is given by A=πr2A = \pi r^2 and the circumference is given by C=2πrC = 2\pi r, where rr is the radius. We are given that the area is 16π16\pi, so 16π=πr216\pi = \pi r^2 r2=16r^2 = 16 r=4r = 4

The circumference is given as 8π8\pi, so 8π=2πr8\pi = 2\pi r r=8π2πr = \frac{8\pi}{2\pi} r=4r = 4

Step 2: Ratio of central angle to total interior angle

The total interior angle measure of a circle is 2π2\pi radians or 360360^\circ. The central angle is given as θ=π3\theta = \frac{\pi}{3} radians. The ratio of the central angle measure to the total interior angle measure is: θ2π=π32π=π312π=16\frac{\theta}{2\pi} = \frac{\frac{\pi}{3}}{2\pi} = \frac{\pi}{3} \cdot \frac{1}{2\pi} = \frac{1}{6}

Final Answer

The ratio of the central angle measure to the total interior angle measure is 16\boxed{\frac{1}{6}}.

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