Radians

Define the radian measure of the angle as the constant of proportionality

So where are we now. We know that for any circle,

• • s = Θr

This would be really useful, but we have no idea what θ is. But we can figure this out. Here are the key things to remember whenever you're doing these problems

• • An angle is a part of a full rotation

    • a full rotation would be 360°

  • An arc is a part of a full circumference

    • a full circumference is 2πr (circumference = 2 x pi x radius)

Since an angle is only a fraction of a rotation, we can only use that fraction of the circumference. Let's look at an example.

Here we have an arc S with subtending angle 30° and a radius of 24 inches. We want to know the length of S. First let's find the circumference of the whole circle, just like in 7th grade.

C = 2 π R

= 2 x π x 24 in

= 48π in

But S isn't the whole circumference, it's only a fraction of it. So now we need to figure out what that fraction is.

To find the fraction, we need to look at our angle. We ask, what fraction is 30° of a whole circle (360°)? Remember, a fraction means part over whole

For our angles, our part is 30° and the whole circle would be 360°. So the fraction of circle we're working with is

= =

So we're only working with 1/12 of a circle.

Well now that we know that, we know we must also use 1/12 of the circumference. Since we already calculated that our circumference is 48π in, we just need to multiply it by 1/12

S = x C

S = x 48π in

S = 4π in

This is a great way to think about it, and I recommend you calculate arcs in this style. However, eventually, you may want to have a formula to do everything for you. We know that arc length is a fraction of the circumference, so let's start here

S = x C

We found that it's easy to find this part over whole by using the angle.

Well if we want a formula, we need to use a variable for the angle. We'll call it m. On the other hand, a full circle is always 360°, so we can leave it like that. Also C = 2 π r so we can substitute

S = x 2 π R

S = R

So here's our formula for arc length. But something looks familiar here. Didn't we have an equation earlier that looked like this?

S = θR

S = R

See the similarity? Does this mean that θ = ? It does. The proportionality constant between arc length and radius is , or θ for short. But this is an important proportionality constant, so it has it's own name

• • θ = , also known as the radian measure.

(It's called a radian measure because is actually a formula for converting degrees into a different unit for measuring angles, called radians. They are way better than degrees for doing math. You'll learn more about these next year.)