Question

If the measure of EF is 90° , what is the arc length of EF?

Answer 1

`\huge\boxed{15 \pi \ \text{or} \approx 47.1 \ \text{in.}}`

Explanation:

We can note a couple of relationships in this circle.

The arc length will be a fraction of the circumference. It will be the same fraction of the circumference that the central angle is to the entire circle.

First step: Find the circumference of the circle.

The circumference of any circle can be defined by the formula
`2 \pi r`, where r is the radius of the circle. The radius is given to us, 30 in. We can now substitute that into the formula.

• 
  `2\cdot \pi \cdot 30`
• 
  `60 \cdot \pi`
• 
  `60\pi`

So our circumference is 60π.

Second Step: Find the ratio of the central angle of the arc to the total circle degrees

We know that the total amount of degrees in a circle is 360°. Therefore, we can set up a proportion to find the ratio between the central angle (90°) and the total circle measurement.

`(90)/(360)`

Third Step: Equal out the two proportions and solve for the missing arc length

Now that we have our base proportion (
`(90)/(360)`), we can turn 60π into a proportion as well, leaving 60π as the denominator so we can solve for the arc length.

`(x)/(60 \pi) = (90)/(360)`

We can now solve for x by cross multiplying.

• 
  `(90)/(360) = (1)/(4)`
• 
  `(x)/(60\pi) = (1)/(4)`
• 
  `x = (60\pi \cdot 1)/(4)`
• 
  `x = (60\pi)/(4)`

• 
  `x = 15\pi \approx47.1`

Hope this helped!