# Lesson Explainer: Angles in Degrees, Minutes, and Seconds Mathematics

In this explainer, we will learn how to convert the measure of angles from degrees, minutes, and seconds to only degrees and vice versa.

Angles are usually formed by two rays that share one of their endpoints, called the vertex of the angle. The two rays are called the sides of the angle. The measure of the angle is a measure of the space between the two sides of the angle.

We can measure an angle by referring to the rotation that takes one side of the angle to the other side. We then have angle measures that correspond to, for instance, a quarter turn or a half turn.

When using degrees to measure angles, a full turn corresponds to an angle of , a quarter turn then corresponds to an angle of , a half turn corresponds to an angle of , and so on.

The measure of an angle is not necessarily a whole number of degrees. In this case, we can of course use a decimal to express the measure of the angle in degrees. We can also use two subunits of degrees: minutes and seconds. Note the analogy with the time units minutes and seconds!

### Definition: Minutes and Seconds

A degree () is made of 60 minutes (), and a minute is made of 60 seconds ():

Before looking into more details at how to convert decimal degrees to degrees, minutes, and seconds, let us look with two examples at how we can do this conversion with a scientific calculator.

### Example 1: Converting an Angle Expressed in Only Degrees to Degrees, Minutes, and Seconds Using a Calculator

Using a calculator, write in degrees, minutes, and seconds.

To convert into degrees, minutes, and seconds, we need to enter 18.15 in the calculator. Then, it depends on the calculator we are using. Often, we have to use the key ; either we use it just after having entered 18.15 and then we press , or we press first, then , and then . Then, what we will see on our calculator is

So, , which reads “18 degrees, 9 minutes, and 0 seconds.

Let us now look at a conversion from degrees, minutes, and seconds to only degrees using the calculator.

### Example 2: Converting an Angle Expressed in Degrees, Minutes, and Seconds to Only Degrees Using a Calculator

Using a calculator, write in degrees.

We need to enter into our calculator. This depends on the calculator we are using. It is often done by pressing the key after the number of degrees, of minutes, and of seconds, followed by and then :

We get

So, .

Let us now use our understanding of the definition of minutes and seconds to do these conversions by hand.

We can illustrate the fact that 1 degree is 60 minutes with a double number line:

So, if we consider, for instance, 36 minutes, it is of 1 degree; that is, . We find the decimal degree by dividing the number of minutes by 60.

We have the same relationship between minutes and seconds, and we can illustrate the relationship between degrees, minutes, and seconds with a triple number line:

### How To: Converting Minutes and Seconds to Decimal Degrees

To convert minutes to decimal degrees, we divide the number of minutes, , by 60: .

To convert seconds to decimal degrees, we divide the number of seconds, , by 3‎ ‎600: .

Let us use this to convert an angle measure in degrees, minutes, and seconds to decimal degrees.

### Example 3: Understanding How to Convert an Angle Expressed in Degrees, Minutes, and Seconds to Only Degrees without Using a Calculator

Fady is trying to convert to degrees only without using a calculator. First, he converts the minutes to degrees by dividing 47 by 60, and then he converts the seconds to minutes by dividing 35 by 60. Finally, he adds all of the parts of the degree to get his answer. His answer is .

1. Is his process correct?
1. Yes
2. No
2. If you think that his process is wrong, which of the following is correct?
1. He should divide the 35 seconds by 60 times 60(3‎ ‎600) to convert it to degrees. So, his answer will be .
2. He should add all of the degrees, minutes, and seconds by dividing the minutes or seconds by 60. So, his answer will be .
3. I think this process is correct.

Part 1

We are considering in this question the measure of an angle of . Remember that and . Therefore, are of 1 degree, and Fady is correct to convert the minutes to degrees by dividing 47 by 60.

However, Fady uses the same procedure to convert the seconds to minutes, that is, by dividing 35 by 60. As , dividing by 60 gives the number of minutes, not that of degrees. Therefore, Fady’s process is wrong (option B).

Part 2

As shown in the following diagram, we have and , so . We need to divide by to convert to degrees.

We find that

Hence,

Option A is the correct answer.

### Example 4: Converting an Angle Expressed in Degrees, Minutes, and Seconds to Only Degrees without Using a Calculator

Without using a calculator, write in degrees.

We consider here the angle of measure . As , is of (), that is, half a degree, so .

As , we convert seconds to minutes by dividing by 60. If we divide again by 60, we convert these minutes to degrees. Therefore, we have

Finally, we get

We find that .

If now we have an angle measured in decimal degrees, we can convert this measure to degrees, minutes, and seconds using the fact that .

Let us take, for instance, . It is plus a quarter of a degree. As a degree is , a quarter of a degree is . Therefore, .

The decimal part of 2.25, 0.25, gives us the fraction of the degree in addition to the 2 whole degrees. Here, we easily recognize that , but we can express any decimal part as a fraction. For instance, . Finding of allows us to convert to minutes:

We see that does not correspond to a whole number of minutes. We can now convert to seconds exactly in the same way as we converted a decimal degree to minutes:

We find that

### How To: Converting Decimal Degrees to Minutes and Seconds

1. The whole part of the measure of an angle in decimal degrees is the whole number of degrees.
2. Multiplying the decimal part by 60 gives the number of minutes.
3. If this number of minutes has a decimal part, then multiplying this decimal part by 60 gives the number of seconds.

Let us use this method in our final example.

### Example 5: Converting an Angle Expressed in Only Degrees to Degrees, Minutes, and Seconds without Using a Calculator

Without using a calculator, write in degrees, minutes, and seconds.

We consider here the measure of an angle of .

The whole part is .

The decimal part is ; by multiplying it by 60, we find the number of minutes:

As is a whole number, there are zero seconds.

Hence,

Let us summarize what we have learned in this explainer.

### Key Points

• A degree () is made of 60 minutes (), and a minute is made of 60 seconds (): ; .
• To convert minutes to decimal degrees, we divide the number of minutes by 60: .
• To convert seconds to decimal degrees, we divide the number of seconds by 3‎ ‎600: .
• To convert decimal degrees to minutes and seconds,
• the whole part of the measure in decimal degrees is the whole number of degrees;
• multiplying the decimal part by 60 gives the number of minutes;
• if this number of minutes has a decimal part, then multiplying this decimal part by 60 gives the number of seconds.