What is the length of a minor arc that is intercepted by a central angle whose measure is 50 degrees and the circle has a radius of 10?

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As you know, the length of an entire circumference of a radius `r` is `2 pi r,` and an entire circumference contains the central angle of 360 degrees.

Also the length of an arc is proportional to the measure of its central angle. The proportion is:

length of an arc with an angle `alpha` : length of an entire circumference = `alpha` : 360°,

or

length for angle alpha : `2 pi r` = `alpha` : 360°

(alpha is in degrees also).

 

Thus the length is `2 pi r * alpha/360,`  in our case it is  `(2 pi *10*50)/360=pi*25/9,` or approximately `8.73.`

 

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We are given a central angle of 50 degrees and a radius of 10, and we are asked to find the length of the minor arc that is subtended:

The length of the arc is a portion of the circumference; specifically it is 50/360 of the circumference.

So to find the length of the minor arc, we multiply the circumference by 50/360.

`s=50/360 * 2*pi*10=(25 pi)/9~~8.73 `

(Note that the circumference is 20pi or approximately 62.83 inches; the arc is 5/36 of this or roughly 1/6 so the answer is reasonable.)

The length of the minor arc is ` (25 pi)/9~~8.73 ` inches.

Approved by eNotes Editorial Team

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