直观感受：通过弧度里的弧长可以得到角度

(Video) How to Measure Angles Using a Protractor

by Eric Burton

The above video by Erin Burton shows you how to use a protractor to determine the measure of an angle. Notice how protractors are circular: that's because circles and rotations are a great way to think about angle measures.

Now that we've learned what an angle is, we want to know how to measure them.

Food for thought: what's the shape of your protractor?

Key idea: angles measure "rotations"

Since angles are closely related to "rotations" (which we'll study more carefully later, the circle will be a helpful tool, because intuitively, it has more "rotational symmetry" than any other object (if you spin it around its center, it "stays put").

Intuitively (and you can already measure this with your protractor if you'd like), the angles $\angle{AOB}$, $\angle{AOY}$, $\angle{XOY}$, $\angle{XOB}$ should all have the same "measure": it takes just as much "angle" to rotate (ray) $OA$ to (ray) $OB$ as it does to rotate $OA$ to $OY$ as it does to rotate $OX$ to $OY$ as it does to rotate $OX$ to $OB$.

(By the way, the angles $\angle{AOB}, \angle{XOY}$ (also $\angle{BOA}$ or $\angle{YOX}$) are called central angles, but don't worry too much about this. The important thing is that $O$ is a "center of rotation".)

Focusing on segments of the same distance from the "center of rotation" (the "axles of a wheel")

So, as hinted above, it turns out that the easiest way to measure that "angle"—that "how much it takes to rotate one ray to another"—is to focus on segments of the same length. So to measure the "angle" it takes to "move" (rotate) ray $OXA$ to ray $OYB$, we'll look at the movement along the circle centered at $O$ with fixed radius $OA = OP = OQ = OR = OS = OB$. (You can imagine a counterclockwise "sweeping motion", rotating $OA$ to $OP$ to $OQ$ to $OR$ to $OS$ to $OB$.)

What's the length of an arc? (and some arc terminology)

Somehow then, the arc $APQRSB$ (the arc containing points $A,P,Q,R,S,B$) is very closely related to the angle $\angle{AOB}$. So we'll later define angles in terms of arc length. (Food for thought: isn't the green curve/mark an arc of a circle itself?.)

First, some arc vocabulary: a semicircle is a "half"-circle (for example, diameters $AM$ and $BN$ both split the circle into $2$ semicircles). The easiest way to name an arc is by writing three or more points on it in order. For example, arcs $APQRSB$, $APB$, $ARSB$, etc. all describe the arc corresponding to the green angle; $ABM$ describes the arc corresponding to the orange angle (note that the arc forms a semicircle); and $ABMN$ or $AMN$ or $ABN$ describes the arc corresponding to the blue angle.

Why do we need three or more points? Well, if we just said "arc $AB$", then we could either be referring to the "big" yellow arc $BMNA$, or the "little" green arc $APQRSB$. On the other hand, there is something to "big" and "little": we can call the "big" arc $BMNA$ the major arc $AB$, because it is larger than a half-circle-arc (it's bigger than the the orange angle!). Similarly, we can call the "little" arc $APB$ the minor arc $AB$, because it is smaller than a half-circle-arc (it's smaller than the orange angle!).

Defining arc length is actually a little tricky to do formally, but you probably have really good intuition for the arc length already: for example, if you lay a string along the arc $APQRSB$, and then measure the result on a ruler, that's the arc length.

That's it! If you want a taste of how we could "rigorously define arc length", the basic idea is that we can approximate it by a bunch of line segments, as well as we'd like (the orange path is a bad approximation, the blue path is a better approximation, and we could keep going if we wanted to).

(If you'd like even more rigor, the Foundations map might be the perfect fit for you. Pun not intended! :)

When you draw an angle in a circle, with the center as its vertex, what's so special about it? Nothing, you might think, but that means that you need to generalize more than that it's an angle. Look at the edge, more like an arc, corresponding to the angle. Assuming the circle is a unit circle, the arc is $x$ units long, and therefore the measure of the angle is $x$ radians.

Text: What is arc measure?

It is important to note that arc measure and arc length are two different concepts. Arc measure is specifically the degree measure of the central angle that defines the arc, while arc length is the length of the arc itself as a segment of the circumference.

A useful consequence of the difference between the two concepts is that you can relate them to one another, or even use them together to find other properties of a circle - like radius, circumference, and area! Read more about properties of circles here.

For example, we know that the two values are both by definition proportional like so: $$\frac{arc\quad measure}{360^\circ}=\frac{arc\quad length}{circumference}$$

Thus, we can find the circumference (and other resulting properties of the circle) through some basic manipulation of the expression above.

Angles don't care about length or "size"—just the "shape" at the vertex

An angle's measure tells you how close two paths are to each other. But angles don't care about the length or "size" of the paths—just the "shape" at the vertex.

That's why if you and your friend have protractors of different sizes, you'll still get the same answer when measuring angles!

Here's another analogy. If you cut an angular slice of cake, with the angle's vertex at the cake's center, then the angle measure wouldn't really represent the size of the slice—just the fraction cut out.