Angles and Their Measurements

- Trigonometry Angles

What are angles? How are they measured? Answering these questions is the focus of this article, part of a series on trigonometry. This article will identify all the constituants of an angle, describe several special angles, and show how angles have been measured for thousands of years.

The first step to understanding angles, is understanding what a ray is. Rays are defined by a point on a fixed line, and all the points to one side, or the other, of that point.

Simple ray $ \overrightarrow{AB} $

Angles then, are two arrays that share a common starting point, known as a vertex.

Simple angle

\[\angle{A} = \angle{BAC} = \angle{CAB}\]

Angles are also commonly given Greek names.

Angles are more clearly described as a fixed ray, known as the initial side, and a second ray, the terminal side, which is rotated away from the first.

Angle sides

An angle with a vertex at the origin of an x,y coordinate system, and at least one ray on the positive x-axis, is in standard position.

Angle standard position

Angles with their vertex at the center of a circle is known as a central angles.

Central angle

Angles can be measured by the rotation of the terminal side, from the initial side. The vertex is placed at the center of a circle, which is divided into 360 equal arcs. One arc is equal to 1 degree, or $ 1° $. An intercept arc is the arc between the initial side and the terminal side.

Angle degrees circle

There are a few types of angles special enough to be given a name, these are the following.

Coterminal Angles

Suppose there is an angle, like the one shown in Figure 6. There are two ways to measure this angle: by rotating from the initial side clockwise, $-90°$ from the initial side, and by rotating $270°$ from the initial side, moving counter-clockwise. The total distance between these two rotations, $-90°$ and $270°$, is $360°$. These angles both share the same terminal sides, which makes them coterminal. Any further complete, $360°$ rotations would also be coterminal angles.

This property can be expressed in the following formula, which states that angle $\alpha$ and $\beta$ are only coterminal there is a $k$ where

\[ m\left(\beta\right) = m\left(\alpha\right) + k360 \]

Degrees, Minutes, and Seconds

Degrees can be divided into 60 equal parts, called minutes. In turn, each minute can be divided int 60 equal parts, known as seconds. If this sounds like a clock on the wall, that’s because it effectively is. Clocks are simply circles that have been divided into 60 equal parts.

An example angle using the degrees-minutes-seconds format would look something like the following:

\[ 37°18'46'' \]

This is read as 37 degrees, 18 minutes, and 46 six seconds. This is much like reading time, only with degrees filling in for hours.

Converting Degrees-Minutes-Seconds to Decimal Degrees

Working with angle measurements in degrees-minutes-seconds format can be a real drag when using a calculator. Sure, most graphing calculators have utilities for entering fractional degrees in this way, but its cumbersome at best. This, then, is a motivation for being able to convert degrees-minutes-seconds to decimal degrees.

Using the previous example, the goal will be to convert $ 37°18'46’’ $ to decimal degrees. Before digging in though, its best to take stock of what is already known about the relationship between degrees and minutes, and degrees and seconds.

\[ 1\text{ degree} = 60\text{ minutes} \] \[ 1\text{ degree} = 3600\text{ seconds} \]

As with other relations, these can be written as fractions. In the example of the degree to minute ratio, these can be

\[ \frac{1\text{ deg}}{60\text{ min}}\text{ or }\frac{60\text{ min}}{1\text{ deg}} \]

If there are two ways to write the equivalent fractions, how is it possible to know which one is the right one to use?

In cases like this, it is important to remember the goal. In this case the goal is turning 18 minutes and 46 seconds into fractions of a degree. Start with converting 18 minutes

\[ \frac{18\text{ min}}{1}*\text{??} = \text{ ?? deg} \]

Looking at the problem this way, in terms of what is needed to find the solution, makes selecting the correct conversion ration easy. Pick the one that will allow the minute units to cancel


\[\frac{18\ \cancel{\text{min}}}{1} * \frac{1\ \text{deg}}{60\ \cancel{\text{min}}} = \frac{18}{60}\text{ deg} = \frac{3}{10}\text{ deg} \]

Now to convert the seconds:

\[\frac{46\ \cancel{\text{sec}}}{1} * \frac{1\ \text{deg}}{3600\ \cancel{\text{sec}}} = \frac{46}{3600}\text{ deg} = \frac{23}{1800}\text{ deg} \]

Finally, the minutes and seconds units have been converted to degrees, so the decimal degrees can be found by adding these constituant pieces together.

\[ 37°18'46'' = 37 + \frac{3}{10} + \frac{23}{1800} \approx 37.3128° \]

Converting Decimal Degrees to Degrees-Minutes-Seconds

Going the reverse direction is fairly simple. First, find the minutes in the fractional degrees and then, if any fractional minutes exist, find the seconds.

Taking the answer found in the previous section, it is possible to go the reverse direction. *Note: since the previous section only found an approximate answer, the result of converting back will also be, at best, an approximate of the original.

\[\frac{.3128\ \cancel{\text{deg}}}{1} * \frac{60\ \text{min}}{1\ \cancel{\text{deg}}} = 18.768\ \text{min}\] \[\frac{.768\ \cancel{\text{min}}}{1} * \frac{60\ \text{sec}}{1\ \cancel{\text{min}}} = 46.08\ \text{sec}\]

This gives $ 37°18'46.08’’ $, right within the ballpark of the original before conversions.


In this article, angles have been introduced as well as defining the parts of an angle, special types of angles, and finally, how angles are measured. The next article will dive into the world of radians, a closer look at the arc length, and the area of a circle defined by an angle.