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.
