Main content

### Course: High school geometry > Unit 8

Lesson 3: Arc length (from degrees)# Subtended angle from arc length

Watch Sal solve an example where he finds the central angle given arc length. Created by Sal Khan.

## Want to join the conversation?

- How did the 18 go from the top to the bottom(17 votes)
- Dividing by 20𝜋 is the same as multiplying by 1∕(20𝜋).

Also, we can write 𝜋 as 𝜋∕1

So, (221∕18)𝜋∕(20𝜋) = 221∕18⋅𝜋∕1⋅1∕(20𝜋)

Now we have a product of fractions, which we know is the same as the product of the numerators divided by the product of the denominators.

221∕18⋅𝜋∕1⋅1∕(20𝜋) = (221⋅𝜋⋅1)∕(18⋅1⋅20𝜋)(15 votes)

- Hello, I have been working diligently on the circle/arc module, and have gotten to the test area, where I am encountering a few problems--after revealing the solution, and plugging the values in sundry ways, still I am not getting this equation, please help -

θ/360 = s / c

θ/360 = 5/6π/20π

θ = 15

-----------

the solution says this should be 15 degrees, and I keep getting 18, please can someone explain the steps exactly. thanx so much.

j(13 votes)- θ/360=5/6pi/20pi

So θ/360=5/120

θ=3*5=15(3 votes)

- what would be some examples of why we'd need to know this? Or what profession rather would someone use this sort of geometry? Please and thanks(9 votes)
- a draftsman\designer would use degrees and the machinist would read the print in degrees to make the part. Any tradesman that fabricates things would use degrees. Radians are left to the sciences.(6 votes)

- At1:06in the video Sal said " in terms of 2 pi radians around the circle," What are radians?(6 votes)
- Radian is one of the unit to measure angles.

1 radian = 180/pi degrees

2pi radians= 360 degrees

1 degree = pi/180 radians

To know more about it see the videos on following link :

https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/radians_tutorial/v/introduction-to-radians(12 votes)

- I don't quite understand the process when you put everything over each other and solve. Is there a simpler way of thinking about it or solving it?(9 votes)
- If two chords of a circle are equal , than their corresponding arcs are equal?(4 votes)
- Yes, if the chords are the same length on the same size circle.(3 votes)

- I'm really having a hard time wrapping my head around this. How exactly does he do that?(2 votes)
- In this example, it is given to us that the circumference is 20pi. This means the perimeter of the circle is 20pi. The arc around the angle we have to find (can call it theta in this case) is 221pi/18. A full circle is 360 degrees. 221pi/18 is a part or fraction of the entire circle (20pi in this case) and theta is a fraction of the entire circle (360 degrees). Since the arc opposite to the central angle of a circle is equal to a central angle, we can set up a proportion. We cannot say the angle theta is 221pi/18 (this is true) because the questions wants our answer in degrees and anything with pi in it will be radians (another way to measure angles revolving around pi). The proportion will be ->

(221pi/18)/20pi = theta/360

now we can solve this using algebra.

1st multiply both sides by 360 since we need to find theta and by multiplying my 360 you isolate theta on one side. If we were to distribute this we get -

360(221)(pi)/18(20)(pi)

now the pi's cancel out and 360/20 = 18 so the 20 is gone, and then we have two 18's so we can cancel those out as well and we are left with 221/1 = 221 degrees!

I really hope this was helpful, if you have nay questions make sure to ask I am happy to help(6 votes)

- I was just wondering if theta was just another way of replacing an unknown value (variable).(3 votes)
- Yes, Greek letters are commonly used for variables to represent an unknown/solvable angle.(4 votes)

- what would be some examples of why we'd need to know this? Or what profession rather would someone use this sort of geometry? Please and thanks(4 votes)
- Can arc length and arc measure be negative?(4 votes)

## Video transcript

A circle has a
circumference of 20 pi. It has an arc
length of 221/18 pi. What is the central angle
of the arc in degrees? So they're asking for this one. So this is the arc that
they're talking about, that's 221/18 pi long. And they want to know this
angle that it subtends, this central angle
right over here. So we just have to
remind ourselves that the ratio of
this arc length to the entire
circumference-- let me write that down-- the
ratio of this arc length, which is 221/18 pi, to the
entire circumference, which is 20 pi, is going to
be equal to the ratio of this central angle,
which we can call theta, the ratio of theta
to 360 degrees if we were to go all the
way around the circle. This will give us
our theta in degrees. If we wanted it in radians,
we would think of it in terms of 2 pi radians
around the circle, but it was 360 degrees
since we're in degrees. Now we just have to simplify. Now the easiest thing
is just to multiply both sides times 360 degrees. So let's do that. So if we multiply the
left-hand side by 360 degrees, we get 360 times 221
times pi over-- let's see, we have 18 times 20 times pi. And on the right-hand side,
if we multiply it by 360, we are just left with theta. So we really just have
to simplify this now. Pi divided by pi
is going to be 1. 360 divided by 20,
well, it's going to be the same thing as 36/2,
which is the same thing as 18. And 18 divided by 18 is 1. So this all simplified
to 221 degrees. Theta is 221 degrees.