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This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ times the radius, a full circular rotation is $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians. So
See [link] . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.
An arc length $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.
This ratio, called the radian measure , is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ to the radius $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ See [link] .
If $\text{\hspace{0.17em}}s=r,$ then $\text{\hspace{0.17em}}\theta =\frac{r}{r}=\text{1radian}\text{.}$
To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is $C=2\pi r,$ where $r$ is the radius. The smaller circle then has circumference $2\pi (2)=4\pi $ and the larger has circumference $2\pi (3)=6\pi .$ Now we draw a 45° angle on the two circles, as in [link] .
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.
Since both ratios are $\text{\hspace{0.17em}}\frac{1}{4}\pi ,$ the angle measures of both circles are the same, even though the arc length and radius differ.
One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360°) equals $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians. A half revolution (180°) is equivalent to $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ radians.
The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ is the length of an arc of a circle, and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is the radius of the circle, then the central angle containing that arc measures $\text{\hspace{0.17em}}\frac{s}{r}\text{\hspace{0.17em}}$ radians. In a circle of radius 1, the radian measure corresponds to the length of the arc.
A measure of 1 radian looks to be about 60°. Is that correct?
Yes. It is approximately 57.3°. Because $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians equals 360°, $1\text{\hspace{0.17em}}$ radian equals $\text{\hspace{0.17em}}\frac{\mathrm{360\xb0}}{2\pi}\approx \mathrm{57.3\xb0}.$
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