A circle has a circumference of
16𝜋 units. Find, in degrees, the measure of
the central angle of an arc with a length of three 𝜋 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, 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 units,
and the circumference is 16𝜋 units. If we let our central angle be 𝜃,
we have 𝜃 over 360 is equal to three 𝜋 over 16𝜋.
We can simplify a little bit. The 𝜋 in the numerator and the 𝜋
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 𝜃 is equal to three
times 360 degrees divided by 16. When we do that, we get 67.5