Question Video: Calculating the Measure of an Arc in Degrees given Its Length and Radius

An arc on a circle with a radius of 50 has a length of 115. Determine the arc’s measure to the nearest tenth of a degree.

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Video Transcript

An arc on a circle with a radius of 50 has a length of 115. Determine the arc’s measure to the nearest tenth of a degree.

Let’s summarize what we know on a quick sketch of this problem. We have circle with a radius of 50 units. An arc which is part of the circumference of this circle has a length of 115 units. We’re then asked to determine the arc’s measure, which we recall is equal to the central angle of the sector. We can use the Greek letter 𝜃 to represent this angle. Now, we recall that the formula for calculating an arc length is 𝜃 over 360 multiplied by 𝜋𝑑. We multiply the circumference of the full circle by the fraction of the circle that we have.

We can substitute the values we know into this formula to give an equation. The arc length is 115. We don’t know the value of 𝜃, so we’ll leave that as it is. And the radius of the circle is 50, which means that the diameter, which is twice the radius, will be 100. We, therefore, have the equation 115 equals 𝜃 over 360 multiplied by 100𝜋. And we can solve this equation to determine the value of 𝜃. First, though, we can simplify if we wish. On the right, the fraction 100 over 360 can be simplified to five over 18 by dividing both the numerator and denominator by 20.

To solve for 𝜃, we need to divide both sides of this equation by five 𝜋 over 18. And we recall that in order to divide by a fraction, we multiply by the reciprocal of that fraction. So, we have 115 multiplied by 18 over five 𝜋 is equal to 𝜃. Again, we can simplify a little if we wish by canceling a factor of five from the numerator and denominator to give 𝜃 equals 23 multiplied by 18 over 𝜋.

Evaluating this on a calculator gives 𝜃 equals 131.78029. And we then need to round this value to the nearest tenth, which gives 131.8 degrees. This gives the central angle of the sector. But remember, by definition, it also gives the measure of the arc. The length of the arc is 115 units. The measure of the arc is 131.8 degrees.

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