**Unformatted text preview: **ar from obvious and leads to a counterintuitive scenario which is explored
in the Exercises. Since the diameter of a circle is twice its radius, we can quickly rearrange the
equation in Deﬁnition 10.1 to get a formula more useful for our purposes, namely:
π= 10.1 Angles and their Measure 601 2π = C
r This tells us that for any circle, the ratio of its circumference to its radius is also always constant;
in this case the constant is 2π . Suppose now we take a portion of the circle, so instead of comparing
the entire circumference C to the radius, we compare some arc measuring s units in length to the
radius, as depicted below. Let θ be the central angle subtended by this arc, that is, an angle
whose vertex is the center of the circle and whose determining rays pass through the endpoints
s
of the arc. Using proportionality arguments, it stands to reason that the ratio r should also be a
constant among all circles, and it is this ratio which deﬁnes the radian measure of an angle. s r
θ
r The radian...

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