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?
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=πr2 and the circumference is given by C=2πr, where r is the radius.
We are given that the area is 16π, so
16π=πr2r2=16r=4
The circumference is given as 8π, so
8π=2πrr=2π8πr=4
Step 2: Ratio of central angle to total interior angle
The total interior angle measure of a circle is 2π radians or 360∘. The central angle is given as θ=3π radians.
The ratio of the central angle measure to the total interior angle measure is:
2πθ=2π3π=3π⋅2π1=61
Final Answer
The ratio of the central angle measure to the total interior angle measure is 61.