Properly defining an angle first requires that we define a ray. A ray consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in Figure 1 can be named as ray EF, or in symbolic form [latex]\overrightarrow{EF}[/latex].

Figure 1

An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in Figure 2 is formed from [latex]\overrightarrow{ED}[/latex] and [latex]\overrightarrow{EF}[/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[/latex], or in symbol form [latex]\angle{DEF}[/latex].

Figure 2

Greek letters are often used as variables for the measure of an angle. The table below is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 2.

Figure 3. Angle theta, shown as [latex]\angle \theta[/latex]

Figure 4

Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in Figure 4.

The following video provides an illustration of angles in standard position.

As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is [latex]\frac{1}{360}[/latex] of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90°.

Figure 5

To formalize our work, we will begin by drawing angles on an x–y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. 

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.

Figure 6

Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that [latex]\frac{90^\circ }{360^\circ }=\frac{1}{4}[/latex]. So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a 360° angle, we calculate that [latex]\frac{360^\circ }{360^\circ }=1[/latex]. So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap.

Since we define an angle in standard position by its terminal side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of 0°, 90°, 180°, 270° or 360°.

Figure 7. Quadrantal angles have a terminal side that lies along an axis. Examples are shown.

Converting Between Degrees and Radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.

The circumference of a circle is [latex]C=2\pi r[/latex]. If we divide both sides of this equation by [latex]r[/latex], we create the ratio of the circumference to the radius, which is always [latex]2\pi[/latex] regardless of the length of the radius. So the circumference of any circle is [latex]2\pi \approx 6.28[/latex] times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure 10.

Figure 10

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 [latex]2\pi[/latex] times the radius, a full circular rotation is [latex]2\pi[/latex] radians. So

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.

Figure 11. 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.

Converting Between Degrees and Degrees Minutes Seconds (DMS)

Using our definition of degree measure, we have that 1° represents the measure of an angle which constitutes   of a revolution. Even though it may be hard to draw, it is nonetheless not difficult to imagine an angle with measure smaller than 1°. There are two ways to subdivide degrees. The first, and most familiar, is decimal degrees. For example, an angle with a measure of 30.5° would represent a rotation halfway between 30° and 31° , or equivalently, of a full rotation. This can be taken to the limit using Calculus so that measures like √2° make sense. The second way to divide degrees is the Degree – Minute – Second (DMS) system. In this system, one degree is divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds. In symbols, we write 1° = 60′ and 1′ = 60′′, from which it follows that 1° = 3600′′.

Relating Arc Lengths to Radius

An arc length [latex]s[/latex] 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 [latex]s[/latex] to the radius [latex]r[/latex].

If [latex]s=r[/latex], then [latex]\theta =\frac{r}{r}=\text{ 1 radian}\text{.}[/latex]

Figure 12. (a) In an angle of 1 radian, the arc length [latex]s[/latex] equals the radius [latex]r[/latex]. (b) An angle of 2 radians has an arc length [latex]s=2r[/latex]. (c) A full revolution is [latex]2\pi[/latex] 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 [latex]C=2\pi r[/latex], where [latex]r[/latex] is the radius. The smaller circle then has circumference [latex]2\pi \left(2\right)=4\pi[/latex] and the larger has circumference [latex]2\pi \left(3\right)=6\pi[/latex]. Now we draw a 45° angle on the two circles, as in Figure 13.

Figure 13. 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.

Since both ratios are [latex]\frac{1}{4}\pi[/latex], the angle measures of both circles are the same, even though the arc length and radius differ.

Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 12, suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.

Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, [latex]C=2\pi r[/latex], and for the unit circle [latex]C=2\pi[/latex]. These two different ways to rotate around a circle give us a way to convert from degrees to radians.

Identifying Special Angles Measured in Radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure 14. Memorizing these angles will be very useful as we study the properties associated with angles.

Figure 14. Commonly encountered angles measured in degrees

Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 14, which are shown in Figure 15. Be sure you can verify each of these measures.

Figure 15. Commonly encountered angles measured in radians

Converting between Radians and Degrees

Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.

This proportion shows that the measure of angle [latex]\theta[/latex] in degrees divided by 180 equals the measure of angle [latex]\theta[/latex] in radians divided by [latex]\pi .[/latex] Or, phrased another way, degrees is to 180 as radians is to [latex]\pi[/latex].