Degree and Radian Measure

 

Learning Objective(s)

·         Understand radian measure.

·         Convert from degree to radian measure.

·         Convert from radian to degree measure.

 

 

You know that there can be different units of measurement to measure the same thing. For example, length can be measured in feet and meters and temperature can be measured in degrees Celsius and degrees Fahrenheit. We often use formulas to convert between different units of measurement.

 

There are also two ways to measure angles. You know how to measure them in degrees. Now you will learn how to measure them in radians and how to convert between these two measurements. While degree measure is used in everyday activities such as building construction and surveying land, radian measure is used for many calculations, such as the speed and distance traveled by satellites above the Earth’s surface—including the International Space Station! It is important to be able to measure angles in radians as well as in degrees and to be able to convert between the two systems.

 

 

 

In order for us to be able to define radians, it is necessary to introduce the concept of a central angle. A central angle is an angle whose vertex is at the center of a circle. In the circle below, the center is point O, the length of the radius is r, and  is a central angle.

 

 

Notice that  cuts off or determines an arc  that has length s. The radian measure of a central angle, often denoted by the Greek letter theta (), is defined to be the ratio of the arc length to the length of the radius. So the radian measure of  is given by:

 

 

Arc length, s, and radius, r, must be in the same units.

 

 

 

 

 

 

What is the connection between degrees and radians? Let’s see what happens when the arc is the entire circle:

 

 

The circumference of this circle is given by . This is the arc length of the entire circle, or the arc length corresponding to a central angle of 360°. What is the radian measure of this angle?

 

 

A central angle of 360° has a radian measure of . That is:

 

 

If you divide both sides by 2, you will obtain:

 

 

We’ll use this to convert from degrees to radians and vice versa.

 

While degrees are always written with a degree symbol (°), radians are usually written without any symbol or unit attached. So, for example,  means the tangent of an angle that measures 3°, while  means the tangent of an angle that measures 3 radians.

 

Make sure that you use the same units of length for the length of the radius r and the arc length s. In the first example, they were both in inches; in the second example, they were both in centimeters. When you divide, those units cancel, and you are left with a unitless number.

 

 

 

 

If you take the definition of radian measure, , and multiply both sides by r, you get the formula for arc length:

 

 

 

 

 

We have used  and  to find the radian measure of the central angle as well as the arc length. But you can also use the arc length formula to find the length of the radius, as seen in the following example.

 

 

 

 

A circle has a radius of 8 inches and central angle AOB makes an arc AB of length 2 feet.   What is the radian measure of angle AOB?   A) B) C) 3 D) 4   Show/Hide Answer A) Incorrect. You probably divided 2 by 8. You need to convert the feet to inches so that you have the same units. The correct answer is 3.   B) Incorrect. You probably correctly converted both measurements to inches, but then divided the radius by the arc length. You need to divide the arc length by the radius. The correct answer is 3.   C) 3 Correct. The arc length is 2 feet or 24 inches. Divide this by the radius, 8 inches, to get 3.   D) 4 Incorrect. You probably divided 8 by 2. You need to convert the feet to inches so that you have the same units. You also need to divide the arc length by the radius. The correct answer is 3.

 

 

 

Earlier we saw that . Divide both sides of this equation by 180. You get:

 

 

You can use this to convert the measure of an angle from degrees to radians. For example, if you wanted to convert 30° to radians, you could multiply both sides by 30:

 

 

To say this another way, you multiply 30° by  to get .

To convert 60° to radians, you multiply 60° by  to get .

In a similar fashion, multiply any degree measure by  to convert it to radians.

 

 

 

 

Here is a chart with the angles that we have converted so far.

 

 

Here is the same information, with some additional angles, in a picture format:

 

 

The same procedure works just as well for negative angles. Remember that a negative angle is just an angle that is oriented clockwise when drawn in standard position. A negative number of degrees becomes a negative number of radians.

 

 

How many radians are there in ?   A) B) C) D)   Show/Hide Answer A) Incorrect. You may have correctly found , but then reduced incorrectly. The correct answer is D.   B) Incorrect. It looks like you divided by 180. You need to multiply by . The correct answer is D.   C) Incorrect. You may have correctly found , but made a mistake while reducing. The correct answer is D.   D) Correct. Multiply by  to get .

 

 

 

Suppose you had angles measured in radians and you wanted to know their degree measure. The procedure is similar to the one going from degrees to radians.

 

Again, you know that . Divide both sides of this equation by . You get:

 

 

If you wanted to convert  radians to degrees, you could multiply both sides of the above equation by :

 

 

 

To state this in another way, we multiplied  by to get 30°. In a similar fashion, multiply any radian measure by  to convert it to degrees.

 

 

 

 

Make sure that your answer has a degree symbol. An angle measure written without a degree symbol is always assumed to be in radians.

 

Here is a chart showing equivalent radian and degree measures for some common angles.

 

 

 

 

 

So 1 radian is a little less than 60°, or a little less than one-sixth of a circle.

 

You can use the same procedure for negative angles.

 

 

What is the equivalent degree measure of  radians written in simplest terms?   A) 150° B) C) D)   Show/Hide Answer A) 150° Incorrect. You probably correctly multiplied by , but you left out the negative sign. The correct answer is .   B) Correct. Multiply by  to get , and then simplify the fraction.   C) Incorrect. You probably correctly multiplied by , but you forgot to simplify. The correct answer is .   D) Incorrect. You multiplied by  instead of . Also, when you write an angle in degrees, you must use a degree symbol, and this answer choice does not have one. The correct answer is .

 

 

 

Angles can be measured in two ways: with degrees and radians. Radian measure of a central angle is defined as the ratio of the arc length to the length of the radius. This definition leads to the arc length formula: .

 

Radians and degrees are connected by the relationship . If you wish to convert from degrees to radians, multiply the degree measure by . If you wish to convert from radians to degrees, multiply the radian measure by .