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Illustration of a circle showing the number of radians in a circle.

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 2 π times the radius, a full circular rotation is 2 π radians. So

2 π  radians = 360 π  radians = 360 2 = 180 1  radian = 180 π 57.3

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.

Illustration of a circle with angle t, radius r, and an arc of r.
The angle t sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.

Relating arc lengths to radius

An arc length     s 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 s to the radius r . See [link] .

s = r θ θ = s r

If s = r , then θ = r r =  1 radian .

Three side by side graphs of circles. First graph has a circle with radius r and arc s, with an equivalence between r and s. The second graph shows a circle with radius r and an arc of length 2r. The third graph shows a circle with a full revolution, showing 6.28 radians.
(a) In an angle of 1 radian, the arc length s equals the radius r . (b) An angle of 2 radians has an arc length s = 2 r . (c) A full revolution is 2 π or about 6.28 radians.

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 π r , where r is the radius. The smaller circle then has circumference 2 π ( 2 ) = 4 π and the larger has circumference 2 π ( 3 ) = 6 π . Now we draw a 45° angle on the two circles, as in [link] .

Graph of a circle with a 45 degree angle and a label for pi/4 radians.
A 45° angle contains one-eighth of the circumference of a circle, regardless of the radius.

Notice what happens if we find the ratio of the arc length divided by the radius of the circle.

Smaller circle:  1 2 π 2 = 1 4 π   Larger circle:  3 4 π 3 = 1 4 π

Since both ratios are 1 4 π , the angle measures of both circles are the same, even though the arc length and radius differ.

Radians

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 2 π radians. A half revolution (180°) is equivalent to π 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 s is the length of an arc of a circle, and r is the radius of the circle, then the central angle containing that arc measures s r 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 2 π radians equals 360°, 1 radian equals 360° 2 π 57.3° .

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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