In this explainer, we will learn how to convert radians to degrees and vice versa.
Radians, like degrees, are a unit of measurement for angles. They are used as an alternative to degrees. Let us formally define what we mean by a radian.
We start with a circle of center and radius .
Now imagine that we take another length of the radius , which we curve along the circumference of the circle.
We now have a sector of the circle with radius and arc length . The angle at the center of the circle is defined as 1 radian (rad).
One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
Now that we have defined what we mean by one radian, we will demonstrate how to convert between radians and degrees.
We recall that the distance around a circle is its circumference, and its measure is found by multiplying by the diameter, . We can write this circumference, , as
Equivalently, we can also describe the circumference in terms of the radius as
Looking at the sector of the circle above, with an arc length of , we consider how many arcs of length can be drawn on the circumference. As we defined the arc length, , to be the same as the radius, , then this means that the number of arcs that could fit on the circumference would be .
If we sketch a diagram of this, we can recall that a numerical approximation for is . On the diagram, there are 6 complete arcs and a remaining portion of arcs.
Since arcs make up the full circumference of the circle, then there must be sectors making up the circle itself. Since the angle of each sector is 1 radian, we can say that the sum of these angles is radians.
However, we also know that the sum of the angles in a circle, in degrees, is , which means that we can observe that
This gives an ideal conversion between the two units of measurement. Additionally, we can also divide both sides of this equation by 2 or by 4, giving two alternative conversion facts: and
Similarly, taking the equation , we can divide both sides by to find the value of 1 radian in degrees:
Knowing any of these conversion facts will allow us to convert any angle from radians to degrees, or vice versa.
How To: Converting between Radians and Degrees
We will now look at some examples of how to convert angles in degrees to angles in radians.
Example 1: Converting an Angle from Degrees to Radians
Convert the following angle measures from degrees to radians. Give your answers in terms of in their simplest form.
In order to convert an angle in degrees into one in radians, we remember the conversion fact:
In the first part of this question, we need to convert the angle of to radians. We notice that is half of . Therefore, we can divide both sides of the conversion by 2, which gives
We can, therefore, give the solution that in radians is
Note that, because we need to give our answer in terms of , we leave our answer in this form, rather than converting to a decimal.
To find the angle of in radians, we take the equation and divide both sides by 6, giving
Alternatively, we could also use the angle measurement we worked out in the first part of the question, that is, and divide both sides by 3, giving
Using either method, we see that in radians is
When we need to convert an angle in degrees that is not a factor of , it is often simplest to first find the measure of in radians.
We know that
So, dividing both sides of this equation by 180 would give
As we need to convert to radians, we can multiply both sides of this equation by 55. This gives us
To simplify the fraction, we divide both the numerator and denominator by the highest common factor of 5, which gives
Therefore, is equivalent to radians.
In the previous question, we saw how to convert the angle of into radians, first by dividing by and then by multiplying it by . The same method can be applied to find any angle measurement.
How To: Converting an Angle from Degrees to Radians
To change any angle in degrees to one in radians, multiply the given angle by :
In the following question, we will consider how to change an angle given in radians into one given in degrees.
Example 2: Converting an Angle from Radians to Degrees
Convert radians to degrees.
To convert an angle from radians to degrees, we remember that
We notice that to convert radians into degrees, we will divide both sides of this equation by 3, which gives
Thus, radians can be written in degrees as .
When we are converting angles that are simple factors of radians or , we can utilize the method we have seen in the previous questions. However, there is an efficient way to convert an angle from radians to degrees in one step.
How To: Converting an Angle from Radians to Degrees
To change any angle in radian measure into degrees, we multiply the given angle by :
We will now see how we can apply this in the following example.
Example 3: Converting an Angle from Radians to Degrees
Convert 0.5 rad to degrees giving the answer to the nearest second.
We recall that to change any angle in radian measure into degrees, we multiply the given angle by :
In this question, we can substitute the given angle, 0.5 radians, for the angle . This gives
Next, we need to convert the angle of degrees in decimal form into one in degrees, minutes, and seconds. To do this, we take the whole part of the answer as degrees. So, gives .
For the minutes, we multiply the remaining decimal by 60, and use the whole part of the answer as the 38 minutes. This gives
For the seconds, we multiply the new remaining decimal by 60 and round the answer to the nearest whole number. This gives us
Hence, we can give the answer that 0.5 rad is equivalent to .
In the following example, we will demonstrate an application of these conversion formulas in a more complex example.
Example 4: Solving a Problem Involving Degrees and Radians
Find the value of the two angles in degrees given that their sum is and their difference is radians. Give your answer to the nearest degree.
We begin this question by defining the two unknown angles as and . We are given that their sum in degrees is and their difference in radians is radians. We can then form two different equations:
We can solve these equations for and if the angles are given in the same units of measurement. Therefore, we can use the conversion between radians and degrees:
We can change either both angles to degrees, or both to radians, but as we are asked for the final answer in degrees, then it is most efficient to change the angle of in radians to one in degrees.
Comparing and , we notice that we must divide through by 6. Therefore,
Alternatively, we could use the rule whereby to change an angle in radians to degrees, we multiply the given angle by . So, to change radians to degrees, we do the following:
Either method demonstrates that , which can then be substituted into the second equation, , to give
We can now solve the pair of simultaneous equations using the substitution or elimination method:
To solve by eliminating the variable first, we add equations (1) and (2) as follows:
To solve this equation for , we divide both sides by 2, which gives
Next, substituting into the first equation, , and rearranging, gives
Therefore, we have found that and , so we can give the value of the two angles in degrees as .
In the following question, we can use our knowledge of converting between radians and degrees to help solve a problem involving the angles in a triangle.
Example 5: Solving a Problem Involving Degrees and Radians
Two angles in a triangle are and radians. Find the value of the third angle giving the answer in radians in terms of .
We can sketch a triangle with the two given angles of and radians.
We will need to find the third angle in this triangle as an angle in radians, so we can define this angle to be radians.
To convert the angle of to radians, we multiply the angle by . This would give
The three angles in the triangle can then be written as radians, radians, and the unknown angle, radians.
To find the unknown angle, we can recall that the angles in a triangle sum to . However, as we are considering the angles in radians here, and as , we can also say that, in radians, the angles in a triangle sum to radians
Using this information, we can form an equation in :
We can rewrite and simplify, giving
Subtracting from both sides gives
The missing angle in the triangle is .
We will now summarize the key points.
- Radians and degrees are both units of measurement of angles.
- One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
- We can convert between degrees and radians using
- Alternatively, we can convert an angle in degrees to one in radians by multiplying the given angle by , and we can convert an angle in radians to one in degrees by multiplying the given angle by .