Trying to understand G-C.5

[moberlin]

I am hoping that you can help me understand the kinds of derivations that are referenced in G-C.5 (Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.) How might one use similarity to show that arc length is proportional to radius length? I know I can show arc length is proportional to radius length given the arc length formula but I suspect that is not the intention. Also, to derive the formula for the area of a sector, what can I assume has already been established? Thanks!

[Bill McCallum]

This follows from the fact that all circles are similar. It’s quite fun to figure out why using the definition of a circle and the definition of a similarity transformation. (I can supply the answer later if you want.)

Now, a similarity transformation with scale factor $k$ transforms any length by a scale factor of $k$, including the arc length. So, just as the radius gets multiplied by $k$, so does the arc length. This means that the ratio between the arc length and the radius stays equivalent no matter what the radius; in other words, the arc length is proportional to the radius.

So this tell us that
$$\text{arc length}=\text{constant}\times \text{radius}.$$
Setting the radius equal to 1 tells us what the constant is: it’s just the arc length for a circle of radius 1, which is exactly the radian measure of the angle.

[moberlin]

Thank you for your response. The one area that is still a little fuzzy for me has to do with the assumption that arc length responds to a dilation like any segment length. I understand that this is true but I am wondering what sense students will make of this. It seems that we need to first develop the concept of arc length. How do you suggest arc length be developed – by approximating it with segments of uniform length and then using an informal limits argument (as one might to develop the concept of the circumference of a circle)? Thanks again!

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