SM2 13.2-7: Finding Minor Arc Length - By Percentage

Added: 2026-05-16

[00:00:00]Hello and welcome. So we're going to find the  length of an arc. We're going to use percentages to find it. So arc length is going along the outer  edge of that circle, right? And for our edge of the circle we have a special name for that thing.  We call it the circumference. So if you know your circumference formula that's 2 pi r. And how we  find the length of an arc — I'm just going to call that L — is that you take your circumference  and you multiply it by the percentage of circle that you have. So here we have a certain  amount of circle. We have the 102 degrees. Now how much of that is from the entire section of  circle? Okay let me rephrase that a little bit to make a little more sense. We know that all  the way around a circle is how many degrees?

[00:00:47]360, right? All the way around is 360. And how  much of the 360 do we have? 102, right? We have 102 degrees of the entire 360 and this is your  percentage of the circle. That's how much that you have there. So all we do is we say okay, 2  pi r, and we know our r, our radius is 20. We're told that it's 20. So then it's 2 pi times 20  times our amount of circle we have 102 over 360.

[00:01:19]And that's it, right? You've now found essentially  your answer. You just got to simplify it down. So 2 times 20 that's 40. All right. So 40  pi times 102 over 360. Now be careful, right? 40 is technically over one that's only  going into the numerator. So 102 times 40, um and that will be, what is  that, 4080 I think, over 360.

[00:01:47]Okay. And that's it. That is a little closer to  your exact answer. We've now put all the numbers together, right? But we're not just looking for an  exact answer. We're looking for a simplified exact answer. So then I would simply simplify by hand.  Go for it. Okay. I'm going to use a calculator.

[00:02:05]Over 360. Now you'll notice I did not include pi  in the calculator, right? If you want a simplified fraction do not include the pi. So if you look  on the right side we'll notice a box over a box.

[00:02:18]Okay. If you press it or unpress it, right? You'll  notice that it goes from decimal to fraction vice versa but as soon as we include the pi? Bam. It will automatically reverts to a decimal and you cannot turn it into a fraction. That's because pi is  an irrational number, so if you want the fraction, leave it out for now. You know it's there. Don't  forget about it, right? But 34 thirds is what we're looking for, alright, so 34/3. And we know pi is in the numerator so it's going to stay  in the numerator, and that is your simplified exact fraction. So 34 pi over 3 and that is your  final answer. Okay. If you wanted that decimal approximation generally third decimal place is  what you want to look at, and you would actually just include the pi if you wanted the decimal.  So 35.604 um seven one six seven four right keeps going. If we wanted the decimal, third decimal  place look to the fourth digit, that's a seven.

[00:03:17]Seven rounds up so your final answer would be  35.605. And that would be it. Thanks for watching.