Question Video: Finding a Central Angle Given the Arc in Term of πœ‹

A circle has a circumference of 16πœ‹ units. Find, in degrees, the measure of the central angle of an arc with a length of 3πœ‹ units.

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Video Transcript

A circle has a circumference of 16πœ‹ units. Find, in degrees, the measure of the central angle of an arc with a length of three πœ‹ units.

To solve this question, we need to think about the relationship between the arc length and circumference and the central angle of that arc. If we take the ratio of the arc length over the circumference, it will be equal to the relationship between the central angle and a full rotation. The full rotation will either be 360 degrees or two πœ‹, depending on whether we’re working in degrees or radians.

In this case, we want to find the angle in degrees. So we’ll substitute 360 degrees for our full rotation. The arc length is three πœ‹ units, and the circumference is 16πœ‹ units. If we let our central angle be πœƒ, we have πœƒ over 360 is equal to three πœ‹ over 16πœ‹.

We can simplify a little bit. The πœ‹ in the numerator and the πœ‹ in the denominator cancel out. We can’t reduce three over 16 any further. So we multiply both sides of the equation by 360 degrees. And we’ll have πœƒ is equal to three times 360 degrees divided by 16. When we do that, we get 67.5 degrees.

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