Lesson Video: Angles in Degrees, Minutes, and Seconds Mathematics

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

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Video Transcript

In this video, we will learn how to convert angles in degrees, minutes, and seconds to degrees only and vice versa. We will begin by looking at the different ways we can measure the size of an angle.

There are several ways to measure the size of an angle, for example, degrees or radians. In this video, we will work with degrees. We know that in a complete circle, there are 360 degrees. Angles can be written as follows: as whole numbers, such as 59 degrees and 172 degrees. If we need a more accurate answer when it is not measured to the nearest degree, we could have 84.5 degrees or 231.75 degrees. 84.5 degrees means that we have 85 full degrees plus five-tenths or half of a degree.

There is an alternative way of writing the second or decimal part of our angle, using minutes and seconds. Each degree is split up into 60 parts, each part being one sixtieth of a degree. These parts are called minutes. In turn, each minute is also split up into 60 parts. These parts are called seconds. The size of an angle could, therefore, be written as 49 degrees, 16 minutes, and 41 seconds. This can be written in shorthand as shown, where the circle is the symbol for degrees, the single quotation mark the symbol for minutes, and the double quotation mark the symbol for seconds. We will now look at some questions where we need to convert from degrees, minutes, and seconds to just degrees and vice versa.

Using a calculator, write 18.15 degrees in degrees, minutes, and seconds.

We recall that one degree is equal to 60 minutes. Our notation for minutes is a single quotation mark. We also recall that one minute is equal to 60 seconds and that the symbol for seconds is a double quotation mark. In this question, we want to write 18.15 degrees in degrees, minutes, and seconds. We can see from this number that there are definitely 18 full degrees. The degree part of our answer will, therefore, be 18. We are, therefore, left with 0.15 degrees.

We need to ask ourselves how many minutes is this. As there are 60 minutes in a degree, to convert from degrees to minutes, we need to multiply the decimal part by 60. In this question, we need to multiply 0.15 by 60. This is equal to nine. Therefore, 0.15 degrees is equal to nine minutes. The minutes part of our answer will be nine. As there was no decimal part to our answer when we multiplied 0.15 by 60, there will be no seconds. This means that 18.15 degrees is equal to 18 degrees, nine minutes, and zero seconds.

It is important to note that we can convert from degrees to degrees, minutes, and seconds using one step on a scientific calculator. We can do this using the button that has the degree symbol, the minute symbol, and the second symbol. In this case, we will type 18.15 followed by the degrees, minutes, and seconds button. Pressing this button at any time converts our answer from a decimal to one in degrees, minutes, and seconds. We then press the equals button, giving us an answer of 18 degrees, nine minutes, and zero seconds.

Using a calculator, write 25 degrees, 30 minutes, and 45 seconds in degrees.

We recall that the single quotation mark for the number in the question stands for minutes and the double quotation mark stands for seconds. We also know that there are 60 minutes in one degree. In turn, a minute is made up of 60 seconds. We need to work out how we can use this information to convert 25 degrees, 30 minutes, and 45 seconds into just degrees. One way is to consider all three parts separately. We know that we have 25 full degrees. As one degree is equal to 60 minutes, one minute will be equal to one sixtieth of a degree. We can, therefore, convert 30 minutes into degrees by multiplying 30 by one sixtieth. This is the same as dividing 30 by 60.

In the same way, one second is equal to one sixtieth of a minute. We can, therefore, convert 45 seconds into degrees by multiplying 45 by one sixtieth and then by one sixtieth again. A quicker way to do this would be to recognize that as 60 multiplied by 60 is 3600, then one second is equal to one three thousand six hundredths of a degree. We can, therefore, convert 45 seconds into degrees by multiplying 45 by one over 3600. This is the same as 45 divided by 3600. 30 divided by 60 is equal to 0.5. 45 divided by 3600 is 0.0125. Adding these two answers together with 25 gives us 25.5125. 25 degrees, 30 minutes, and 45 seconds is equal to 25.5125 degrees.

There is a quicker way of doing this in one step on the calculator. This involves using the degrees, minutes, and seconds button that can be found on the majority of scientific calculators. We begin by inputting the number 25. We then press our button. We input 30 followed by the button again. Finally, we input 45 and the button a third time. When we press the equals button, the number 25 degrees, 30 minutes, and 45 seconds will appear on our calculator display. We then need to press the degrees, minutes, and seconds button a final time, which will give us the answer 25.5125. This is the answer that we calculated earlier in degrees.

Our next question is a word problem in context.

Daniel is trying to convert 81 degrees, 47 minutes, 35 seconds 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 two minutes by dividing 35 by 60. Finally, he adds all of the parts of the degree to get his answer. His answer is 82.3667 degrees. Is his process correct? If you think his process is wrong, which of the following is correct? (A) He should divide 35 seconds by 60 times 60 or 3600 to convert it to degrees. So, his answer will be 81.793 degrees. Is it option (B) he should add all of the degrees, minutes, and seconds. So, his answer will be 163 degrees? Or is it option (C) I think the process is correct?

Let’s begin by looking at the angle we are given: 81 degrees, 47 minutes, and 35 seconds. We recall here our notation for minutes and seconds, one quotation mark and two quotation marks, respectively. We also recall that one degree is equal to 60 minutes. This means that to convert from minutes to degrees, we need to divide by 60. Dividing 47 by 60 is correct in order to convert from minutes to degrees. This is equal to 0.7833 and so on or 0.783 recurring.

We also recall that there are 60 seconds in one minute. This means that we can divide 35 by 60 to convert seconds to minutes. However, we want our answer in degrees only, whereas this would give an answer in minutes. 60 multiplied by 60 is 3600. This means that one degree is equal to 3600 seconds. In order to convert 35 seconds into degrees, we would need to divide 35 by 3600. This means that Daniel’s process is not correct. He should have divided 35 by 3600 instead of by 60 to convert the seconds into degrees. The answer of 82.3667 degrees is incorrect.

The second part of our question wants us to identify the correct process. We have already established that option (C) cannot be the correct answer. As we need to divide 35 by 3600, option (A) looks like it could be correct. 35 divided by 3600 is equal to 0.00972 and so on. In order to calculate 81 degrees, 47 minutes, and 35 seconds in degrees only, we need to add 0.7833, 0.00972, and 81. This gives us an answer of 81.793 and so on degrees. Option (A) is the correct process that Daniel should have followed. We know that option (B) cannot be correct, as he has not converted the minutes and seconds into degrees. 163 degrees is not the same as 81 degrees, 47 minutes, and 35 seconds.

Our final two questions involve conversions without a calculator.

Without using a calculator, write 20 degrees, 30 minutes, 45 seconds in degrees.

We recall that the quotation mark symbols in the question represent minutes and seconds, respectively. There are 60 minutes in one degree. This means that in order to convert from minutes to degrees, we need to divide by 60. Likewise, there are 60 seconds in one minute. 60 multiplied by 60 is 3600. Therefore, this is the number of seconds in one degree. To convert from seconds to degrees, we need to divide by 3600. We need to write the three parts of our original number separately.

We begin with 20 degrees. We then have 30 minutes, which in degrees would be 30 divided by 60 or 30 over 60. We then have 45 seconds, which can be converted into degrees by dividing by 3600. The first fraction simplifies to three over six and then one-half. One-half is equal to 0.5. So, 30 divided by 60 is 0.5. We know that 45 multiplied by eight is 360. This means that 45 multiplied by 80 is 3600. So, our second fraction simplifies to one over 80 or one eightieth. We recall that one-eighth is equal to 0.125. It is half of a quarter. Dividing this by 10 gives us that one eightieth is equal to 0.0125. Adding these three values gives us 20.5125. We can, therefore, conclude that 20 degrees, 30 minutes, and 45 seconds is the same as 20.5125 degrees.

Without using a calculator, write 20.7 degrees in degrees, minutes, and seconds.

20.7 degrees is the same as 20 degrees plus seven-tenths or 0.7 of a degree. We also recall that one degree is equal to 60 minutes. We know that the first part of our answer will be 20 degrees. And we need to convert 0.7 degrees into minutes or minutes and seconds. To convert 0.7 degrees into minutes, we need to multiply it by 60. 0.7 multiplied by 60 is equal to 42. This is because 0.7 multiplied by 10 is seven, and multiplying this by six gives us 42. 0.7 degrees is, therefore, equal to 42 minutes. As this was an exact number of minutes, we will have zero seconds. 20.7 degrees is, therefore, equal to 20 degrees, 42 minutes, and zero seconds.

We will now summarize the key points from this video. We found out that angles can be written in degrees, minutes, and seconds or degrees only. We can convert from one form to the other, using the fact that 60 minutes is one degree and 60 seconds is one minute. Combining these two facts, we know that there are 3600 seconds in one degree. As well as multiplying and dividing by 60, we can use the degrees, minutes, and seconds button on a scientific calculator to convert from one form to the other. This can speed up the process, particularly when dealing with more complicated decimals.

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