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### Course: Geometry (all content) > Unit 14

Lesson 5: Arc length (from radians)# Arc length from subtended angle: radians

Sal finds the length of an arc using the radius and the radian measure of the angle subtended by the arc. Created by Sal Khan.

## Want to join the conversation?

- Why exactly is the degree measure corresponding to the length of its arc?(38 votes)
- That is how radian is defined. One radian is the angle where the arc length equals the radius.

You probably know that the circumference of a circle is 2πr.

And 2π radians = 360°. That is not a coincidence.(32 votes)

- Can't you use proportion to solve this instead of what Sal did?(14 votes)
- Yes you could solve this problem using proportions. angle/ 2pi = arc/circumference

0.4/2pi =?/2pi.5. Sals method helps us think about what a radian actually is by definition...and it is faster.(13 votes)

- So this formula is true right?

, ie, radian = length of arc/radius

so, length of arc = radian * radius

and also, radius= length of arc/radian(11 votes) - What does radii mean?(3 votes)
- You can also say "radiuses," but radii is probably better.(3 votes)

- why did he multiply 0.4 radii by 5(length of radius)? what do you get from that? i m still not sure what radian is...so confused T_T(6 votes)
- He multiplied them in order to get the length of the arc.

A radian is a unit for angles, similar to degrees. However, radians measure the amount of radii that is formed when you draw an arc or a part of a circle. This is based on the fact that the radius can fit 2pi(~6.28) times a circle, or a full rotation. There are 2pi radii in a full circle, or 2pi radians.

This gif should help you understand a radian better: http://upload.wikimedia.org/wikipedia/commons/4/4e/Circle_radians.gif(8 votes)

- How did he get 2 from 0.4 and 5?(0 votes)
- An arch is a structure that spans a space and resolves forces into compressive stress. But that is not important to our context.

An arc is a portion of a circle or other curvilinear shape.

Interestingly enough, most arches are in the shape of an arc.(13 votes)

- Could this method have also worked?

0.4 Radians

1 radian = 180/pi degrees

1 radian * 0.4 = 180/pi * 0.4 = 72/pi degrees

72/pi degrees/ 360 = 1/5 pi (Arc length formula: n/360 * 2 pi r)

Then,

2 pi r = 10 pi, so 1/5 pi * 10 pi = 2 units.

2 units is the answer(5 votes)- Why would you ever want to convert radians to degrees first? The equation for arc length in radians is so much simpler:
`l = θr`

`l = 0.4*5 = 2`

In fact, the equation of arc-length in degrees actually converts the degrees into radians with the expression`2π/360`

(which is the same as`π/180`

). So the answer is yes, that works, but it's entirely ridiculous.(4 votes)

- What exactly is a radian? Other than "unit of measure for angles".(4 votes)
- A radian is the angle marked out by an arc whose length is equal to the radius of the circle.(5 votes)

- Khan is using the words "radii", "radiuseses", and "radians". Before this video I had thought they all mean the same thing; that radians is just a plural term for radius. Now I'm not so sure... could someone please explain their relationship?(3 votes)
- Radii and radiuses mean the same thing: more than one radius. Radian, however, is not the same thing. A radian, simply put, is a unit of measure for angles that is based on the radius of a circle. What this means is that if we imagine taking the length of the radius and wrapping it around a circle, the angle that is formed at the centre of the circle by this arc is equal to 1 Radian.

Hope this helped! Let me know if you have any other questions.(4 votes)

## Video transcript

P is the center
of the green arc. So this is p right over here. The measure of angle
P is 0.4 radians, and the length of the
radius is 5 units. That's this length
right over here. Find the length
of the green arc. So just to kind of
conceptualize this a little bit, P, you can imagine, is the
center of this larger circle. And this angle
right over here that has a measure of 0.4 radians, it
intercepts this green arc right over here. In order to figure this
out-- and actually, I encourage you to pause
this video now and try to think about this
question on your own. How long is this arc,
given the information that they've given us? Well, all we have to
do is remind ourselves what a radian is. One way to think about a radian
is if you look at the arc that the angle intercepts, which
is this green arc, if you think about its length, the
length of this green arc is going to be 0.4 radii. One way to think
about radians is if this angle is
0.4 radians, that means that the arc
that it intercepts is going to be 0.4 radii long. So this length we could
write as 0.4 radii. Or "radiuses," but "radii"
is the proper term. Now, we don't want our
length in terms of radii. We want our length in terms
of whatever units the radius is, these kind of 5 units. Well, we know that each
radius has a length of 5, that our radius of the
circle has a length of 5. So this is going to
be 0.4 radii times 5-- and you know they just call
it units right over here. I'll put it in quotes because
it's kind of a generic term-- 5 units per radii. So the radii cancel out. We're left with just the
units, which is what we want. So 0.4 times 5 is 2. So this is going
to be equal to 2. So just as a
refresher again, when the angle measure
in radians, one way to think about it is the
arc that it intercepts, that's going to be
this many radii long. Well, if each radius
is 5 units, it's going to be 0.4 times
5 units long, or 2.