Length of an arc that subtends a central angle

I have a circle here whose
circumference is 18 pi.

So if we were to measure all
the way around the circle,

we would get 18 pi.

And we also have a
central angle here.

So this is the
center of the circle.

And this central angle
that I'm about to draw

has a measure of 10 degrees.

So this angle right
over here is 10 degrees.

And what I'm curious
about is the length

of the arc that subtends
that central angle.

So what is the length of
what I just did in magenta?

And one way to think about
it, or actually maybe the way

to think about it, is
that the ratio of this arc

length to the entire
circumference--

let me write this
down-- should be

the same as the ratio
of the central angle

to the total number
of angles if you were

to go all the way around the
circle-- so to 360 degrees.

So let's just think about that.

We know the
circumference is 18 pi.

We're looking for
the arc length.

I'm just going to call
that a. a for arc length.

That's what we're going
to try to solve for.

We know that the central
angle is 10 degrees.

So you have 10 degrees
over 360 degrees.

So we could simplify this
by multiplying both sides

by 18 pi.

And we get that our arc
length is equal to-- well,

10/360 is the same
thing as 1/36.

So it's equal to
1/36 times 18 pi,

so it's 18 pi over 36, which
is the same thing as pi/2.

So this arc right
over here is going

to be pi/2, whatever units
we're talking about, long.

Now let's think about
another scenario.

Let's imagine the same circle.

So it's the same circle here.

Our circumference
is still 18 pi.

There are people
having a conference

behind me or something.

That's why you might hear
those mumbling voices.

But this circumference
is also 18 pi.

But now I'm going to make the
central angle an obtuse angle.

So let's say we were to
start right over here.

This is one side of the angle.

I'm going to go and
make a 350 degree angle.

So I'm going to go all
the way around like that.

So this right over here
is a 350 degree angle.

And now I'm curious
about this arc that

subtends this really huge angle.

So now I want to figure out this
arc length-- so all of this.

I want to figure out this
arc length, the arc that

subtends this really obtuse
angle right over here.

Well, same exact logic-- the
ratio between our arc length,

a, and the circumference of
the entire circle, 18 pi,

should be the same as the
ratio between our central angle

that the arc subtends, so
350, over the total number

of degrees in a
circle, over 360.

So multiply both sides by 18 pi.

We get a is equal to-- this
is 35 times 18 over 36 pi.

350 divided by 360 is 35/36.

So this is 35 times
18 times pi over 36.

Well both 36 and 18
are divisible by 18,

so let's divide them both by 18.

And so we are left with 35/2 pi.

Let me just write it
that way-- 35 pi over 2.

Or, if you wanted to
write it as a decimal,

this would be 17.5 pi.

Now does this makes sense?

This right over here,
this other arc length,

when our central
angle was 10 degrees,

this had an arc
length of 0.5 pi.

So when you add these two
together, this arc length

and this arc length,
0.5 plus 17.5,

you get to 18 pi, which was
the circumference, which

makes complete sense because
if you add these angles, 10

degrees and 350 degrees, you
get 360 degrees in a circle.