Length of an Arc

Show why the length of an arc is proportional to the radius

Here we have two circles. Each circle has an angle, a radius , and an arc.

Here's something weird to think about, but it's true.

• • If I ever have two numbers, there's some number I can multiply the first number by to get the second number.

  • For example, if I have 2 and 10, I know i can multiply 2 times 5 to get 10

Let's take that and run with it. For the circle on the left, that means there's some number I can multiply the radius (r) by to get the arc (s). For now we don't what that number is, but we're going to call this number θ (theta). Writing this as an equation, we get:

• • radius x theta = arc

  • s = θr

Now let's multiply both sides of this equation by a random number k

• • ks = kθr

Rearranging without changing anything

• • (ks) = θ(kr)

Wait a second, this looks exactly like a dilation, with a scale factor of k. Could it be? Yes, it could, because we proved already that all circles are similar. That means we can now say that

• • ks will equal some new arc (S)

  • kr will equal some new radius (R)

(I drew a new circle to the right, but it could be any size, since we never said how big k was.)

Now that we know ks = S and kr = R, we can substitute

• • (ks) = θ(kr)

  • S = θR

What's important here is that our number θ is still there. Also since we never picked a number for k, that means this is true for any circle. Remember what we call it when the ratio between two numbers always stays the same? We say those two numbers are proportional. Here, the proportionality constant is θ.

• • Arc length and radius for circles with the same subtending angle are proportional, and the constant of proportionality is θ.

Honestly this proof is not very interesting. You have to understand it for the test, but we'll do a different activity in class that will show you what it really means.

(Could do the proof starting with S/s = R/r, rearrange to S/R = s/r, set both of those equal to a proportionality constant Θ, and simplify. This at least prevents the step of having to randomly multiply by k in hopes that it'll randomly help)