Angle Measurement: Degrees, Minutes, Seconds

Click here to jump to the two degrees, minutes, seconds calculators near the bottom of this page.

There are several ways to measure the size of an angle. One way is to use units of degrees. Radian measure is another way. (Definition of a radian) Here we will work with degrees.

In a complete circle there are three hundred and sixty degrees.

An angle could have a measurement of 35.75 degrees. That is, the size of the angle in this case would be thirty-five full degrees plus seventy-five hundredths, or three fourths, of an additional degree. Notice that here we are expressing the measurement as a decimal number. Using decimal numbers like this one can express angles to any precision - to hundredths of a degree, to thousandths of a degree, and so on.

There is another way to state the size of an angle, one that subdivides a degree using a system different than the decimal number example given above. The degree is divided into sixty parts called minutes. These minutes are further divided into sixty parts called seconds. The words minute and second used in this context have no immediate connection to how those words are usually used as amounts of time.

In a full circle there are 360 degrees.

Each degree is split up into 60 parts, each part being 1/60 of a degree. These parts are called minutes.

Each minute is split up into 60 parts, each part being 1/60 of a minute. These parts are called seconds.

a degree equals 60 minutes, a minute equals 60 seconds

The size of an angle could be stated this way: 40 degrees, 20 minutes, 50 seconds.

There are symbols that are used when stating angles using degrees, minutes, and seconds. Those symbols are show below:

Symbol for degree:

Symbol for minute:

Symbol for second:

So, the angle of 40 degrees, 20 minutes, 50 seconds is usually written this way:

How could you state the above as an angle using common decimal notation? The angle would be this many degrees, (* means times.):

40 + (20 * 1/60) + (50 * 1/60 * 1/60)

That is, we have 40 full degrees, 20 minutes - each 1/60 of a degree, and 50 seconds - each 1/60 of 1/60 of a degree.

Work that out and you will get a decimal number of degrees. It's 40.34722...

Going the other way is a bit more difficult. Suppose we start with 40.3472 degrees. Can we express that in units of degrees, minutes, and seconds?

Well, first of all there are definitely 40 degrees full degrees. That leaves 0.3472 degrees.

So, how many minutes is 0.3472 degrees? Well, how many times can 1/60 go into 0.3472? Here's the same question: What is 60 times 0.3472? It's 20.832. So, there are 20 complete minutes with 0.832 of a minute remaining.

How many seconds are in the last 0.832 minutes. Well, how many times can 1/60 go into 0.832, or what is 60 times 0.832? It's 49.92, or almost 50 seconds.

So, we've figured that 40.3472 degrees is almost exactly equal to 40 degrees, 20 minutes, 50 seconds.

(The only reason we fell a bit short of 50 seconds is that we really used a slightly smaller angle in this second half of the calculation explanation. In the original angle, 40.34722... degrees, the decimal repeats the last digit of 2 infinitely, so, the original angle is a bit bigger than 40.3472.)

Here's a calculator that will take you from angle measurement using degrees, minutes, and seconds to angle measurement expressed as a decimal number.

Make up an angle measurement using degrees, minutes, and seconds. Then, using the above procedures, calculate what that angle measurement would be if it were expressed as a decimal number. Finally, enter your original angle in degrees, minutes, and seconds into this calculator, click the 'Calculate' button, and see if your conversion mathematics was correct. Only enter positive integers into this calculator. That is, do not enter any numbers like 5.5, -5.5, or -5.

Click this button to see the actual JavaScript code that executes in the above program:

Here's the code that runs when you click the above 'Calculate' button on the above Degrees, Minutes, Seconds program. It is shown here as a possible aid for your study and understanding.

You can change the code, if you like, and then click the following 'Reevaluate code' button. The program will then work as per your changes when you click the 'Calculate' button on the above Degrees, Minutes, Seconds program. Of course, your changes, especially random changes, can introduce errors, miscalculations, and browser crashes. If you need to get things back to their original condition, just reload this page using your browser's reload button.

Again, the intention here is to conveniently show you the inner workings of this program so that you understand how the program arrives at the result. Click the 'Code' button again to close this section.

window.onClickCalculateDegMinSec = function()
{
	// Variables.
	var degrees    = 0;
	var minutes    = 0;
	var seconds    = 0;
	var decDegrees = 0.0;


	// Get degrees, minutes, and seconds from user.
	degrees = parseFloat(window.document.degMinSec.degrees.value);
	minutes = parseFloat(window.document.degMinSec.minutes.value);
	seconds = parseFloat(window.document.degMinSec.seconds.value);


	// Is degrees negative?
	if(degrees < 0.0)
	{
		// Yes, degrees is negative.
		// Announce the error and quit.
		
		alert('The degrees should be expressed\nwith a zero or a positive number.');

		return;
	}

	// Is minutes negative?
	if(minutes < 0.0)
	{
		// Yes, minutes is negative.
		// Announce the error and quit.
		
		alert('The minutes should be expressed\nwith a zero or a positive number.');

		return;
	}

	// Is seconds negative?
	if(seconds < 0.0)
	{
		// Yes, seconds is negative.
		// Announce the error and quit.
		
		alert('The seconds should be expressed\nwith a zero or a positive number.');

		return;
	}

	// Is degrees an integer?
	if(degrees != Math.floor(degrees))
	{
		// No, degrees is not an integer.
		// Announce the error and quit.
		
		alert('The degrees should be an integer.');

		return;
	}

	// Is minutes an integer?
	if(minutes != Math.floor(minutes))
	{
		// No, minutes is not an integer.
		// Announce the error and quit.
		
		alert('The minutes should be an integer.');

		return;
	}

	// Is seconds an integer?
	if(seconds != Math.floor(seconds))
	{
		// No, seconds is not an integer.
		// Announce the error and quit.
		
		alert('The seconds should be an integer.');

		return;
	}

	// Is degrees greater than or equal to 360?
	if(degrees >= 360.0)
	{
		// Yes, degrees is greater than or equal to 360.
		// Announce the error and quit.
		
		alert('The degrees should be less than 360.');

		return;
	}

	// Is minutes greater than or equal to 60?
	if(minutes >= 60.0)
	{
		// Yes, minutes is greater than or equal to 60.
		// Announce the error and quit.
		
		alert('The minutes should be less than 60.');

		return;
	}

	// Is seconds greater than or equal to 60?
	if(seconds >= 60.0)
	{
		// Yes, seconds is greater than or equal to 60.
		// Announce the error and quit.
		
		alert('The seconds should be less than 60.');

		return;
	}


	// All is well, let's calculate.
	decDegrees = degrees + (minutes * (1.0 / 60.0)) + (seconds * (1.0 / 3600.0));

	// Output result.
	alert('The decimal number of degrees is ' + decDegrees + '.');
}

Here's another calculator that you can use to check your work going from an angle measurement using a decimal number of degrees to the same angle measurement expressed in terms of degrees, minutes, and seconds. The final result is rounded to the nearest second.

Click this button to see the actual JavaScript code that executes in the above program:

Here's the code that runs when you click the above 'Calculate' button on the above Decimal number of degrees program. It is shown here as a possible aid for your study and understanding.

You can change the code, if you like, and then click the following 'Reevaluate code' button. The program will then work as per your changes when you click the 'Calculate' button on the above Decimal number of degrees program. Of course, your changes, especially random changes, can introduce errors, miscalculations, and browser crashes. If you need to get things back to their original condition, just reload this page using your browser's reload button.

window.onClickCalculateDecDeg = function()
{
	// Variables.
	var degrees     = 0;
	var degreesTemp = 0.0;
	var minutes     = 0;
	var minutesTemp = 0.0;
	var seconds     = 0;
	var secondsTemp = 0.0;

	var message = '';

	var isNegativeAngle;


	// Get decimal number of degrees from user.
	degreesTemp = parseFloat(window.document.decDegrees.decDegrees.value);


	// Assume angle is not negative.
	isNegativeAngle = false;

	// Is the decimal number of degrees negative?
	if(degreesTemp < 0.0)
	{
		// Remember that the decimal number of degrees was entered as a negative number.
		isNegativeAngle = true;

		// Work with the absolute value of the decimal number of degrees.
		degreesTemp = -degreesTemp;

	}

	// Keep the integer value of degrees.
	degrees = Math.floor(degreesTemp);

	
	// Find the decimal value of degrees.
	minutesTemp = degreesTemp - degrees;
	// Convert the decimal value of degrees into number of minutes.
	minutesTemp = 60.0 * minutesTemp;
	// Keep the integer value of minutes.
	minutes = Math.floor(minutesTemp);


	// Find the decimal value of minutes.
	secondsTemp = minutesTemp - minutes;
	// Convert the decimal value of minutes into number of seconds.
	secondsTemp = 60.0 * secondsTemp;
	// Round to the integer value of seconds.
	seconds = Math.round(secondsTemp);
	

	// Create a message about if the angle is negative or a positive.
	if(isNegativeAngle)
	{
		message = "This angle is negative.\n";
	}
	else
	{
		message = "This angle is positive.\n";
	}

	// Append the results to the message.
	message += 'Degrees = ' + degrees + '\n';
	message += 'Minutes = ' + minutes + '\n';
	message += 'Seconds = ' + seconds;

	// Announce the message.
	alert(message);
}