Math problems are never fun, especially when you are asked to convert an angle to radians. It also gets harder if you are not sure how to apply the formula, which is why this article was created for you!
Like every other quantity, angles also have units for measurement. Degrees and radians are two units for measuring angles. There are other units to measure the angles (like gradians and MRADs). The most popular unit for measuring angles that most people are familiar with is the degree is written (°). The subunits of a degree are minutes and seconds. There are 360 degrees, 180 degrees for a half-circle (semi-circle), and 90 degrees for a quarter circle (a right triangle) in a full circle or one complete rotation.
Degrees basically state direction and angle size. Facing North means you are facing the direction of 0 degrees. If you turn towards South, you are facing the direction of 90 degrees. If you come back to North after full rotation, you have turned through 360 degrees. Usually, the anticlockwise direction is considered positive. If you turn towards West from North, the angle will be either -90 degrees or +270 degrees.
In geometry, there is another unit for measuring angles, known as the radian (Rad). Now, why do we need radians when we are already comfortable with angles? Most computation in mathematics involves numbers. Since degrees are not actually numbers, then the radians measure is preferred and often required to solve problems. A good example that’s similar to this concept is using decimals when we have percentages. Although a percentage can be shown with a number followed by a % sign, we convert it to a decimal (or fraction).
A circle contains 360 degrees, which is the equivalent of 2π radians, so 360° and 2π radians represent the numerical values for going “once around” a circle. Therefore, degree and radian can be equated as:
2π = 360° And π = 180°
Hence, from the above equation, we can say, 180 degrees is equal to π radian.
Usually, in general geometry, we consider the measure of the angle in degrees (°). Radian is commonly considered while measuring the angles of trigonometric functions or periodic functions. Radians is always represented in terms of pi, where the value of pi is equal to 22/7 or 3.14.
A degree has its sub-parts also, stated as minutes and seconds. This conversion is the major part of Trigonometry applications.
Degrees x π/180 = Radians
Radians × 180/π = Degrees
360 Degrees = 2π Radians
180 Degrees = π Radians
Table of Contents
The Conversion Process
The value of 180° is equal to π radians. To convert any given angle from the measure of degrees to radians, the value has to be multiplied by π/180.
Where the value of π = 22/7 or 3.14
Sound confusing? Don’t worry, you can easily convert degrees to radians, or from radians to degrees, in just a few easy steps.
Write down the number of degrees you want to convert to radians. Let’s work with a few examples so you really get the concept down. Here are the examples you’ll be working with:
- Example 1: 120°
- Example 2: 30°
- Example 3: 225°
Multiply the number of degrees by π/180. To understand why you have to do this, you should know that 180 degrees constitute π radians. Therefore, 1 degree is equivalent to (π/180) radians. Since you know this, all you have to do is multiply the number of degrees you’re working with by π/180 to convert it to radian terms. You can remove the degree sign since your answer will be in radians anyway. Here’s how to set it up:[4]
- Example 1: 120 x π/180
- Example 2: 30 x π/180
- Example 3: 225 x π/180
Do the math. Simply carry out the multiplication process, by multiplying the number of degrees by π/180. Think of it like multiplying two fractions: the first fraction has the number of degrees in the numerator and “1” in the denominator, and the second fraction has π in the numerator and 180 in the denominator. Here’s how you do the math:
- Example 1: 120 x π/180 = 120π/180
- Example 2: 30 x π/180 = 30π/180
- Example 3: 225 x π/180 = 225π/180
Simplify. Now, you’ve got to put each fraction in lowest terms to get your final answer. Find the largest number that can evenly divide into the numerator and denominator of each fraction and use it to simplify each fraction. The largest number for the first example is 60; for the second, it’s 30, and for the third, it’s 45. But you don’t have to know that right away; you can just experiment by first trying to divide the numerator and denominator by 5, 2, 3, or whatever works. Here’s how you do it:
- Example 1: 120 x π/180 = 120π/180 ÷ 60/60 = 2/3π radians
- Example 2: 30 x π/180 = 30π/180 ÷ 30/30 = 1/6π radians
- Example 3: 225 x π/180 = 225π/180 ÷ 45/45 = 5/4π radians
Write down your answer. To be clear, you can write down what your original angle measure became when converted to radians. Then, you’re all done! Here’s what you do:
- Example 1: 120° = 2/3π radians
- Example 2: 30° = 1/6π radians
- Example 3: 225° = 5/4π radians
Example 4: Convert 90 degrees to radians.
Solution: Given, 90 degrees is the angle
Angle in radian = Angle in degree x (π/180)
= 90 x (π/180)
= π/2
Hence, 90 degrees is equal to π/2 in radian.
Degrees to Radian Formula
To convert degrees to radian, we can use the formula as given below.
Degree x π/180 = Radian
Let us see some examples:
Example 5: Convert 15 degrees to radians.
Solution: Using the formula,
15 x π/180 = π/12
Example 6: Convert 330 degrees to radians.
Solution: Using the formula,
330 x π/180 = 11π/6
Negative Degrees to Radian
The method to convert a negative degree into radian is the same as we have done for positive degrees. Multiply the given value of the angle in degrees by π/180.
Suppose, -180 degrees has to be converted into radian, then,
Radian = (π/180) x (degrees)
Radian = (π/180) x (-180°)
Angle in radian = – π
Degrees to Radians Chart
Let us create the table to convert some of the angles in degree form to radian form.
| Angle in Degrees | Angle in Radians |
|---|---|
| 0° | 0 |
| 30° | π/6 = 0.524 Rad |
| 45° | π/4 = 0.785 Rad |
| 60° | π/3 = 1.047 Rad |
| 90° | π/2 = 1.571 Rad |
| 120° | 2π/3 = 2.094 Rad |
| 150° | 5π/6 = 2.618 Rad |
| 180° | π = 3.14 Rad |
| 210° | 7π/6 = 3.665 Rad |
| 270° | 3π/2 = 4.713 Rad |
| 360° | 2π = 6.283 Rad |
More Examples
Question 7: Convert 200 degrees into radians.
Solution: By the formula, we know;
Angle in radians = Angle in degree × π/180
Thus,
200 degrees in radians = 200 × π/180 = 10π/9 = 3.491 Rad
Question 8: Convert 450 degrees into radians.
Solution: By the formula, we know;
Angle in radians = Angle in degree × π/180
Thus,
450 degrees in radians = 450 × π/180 = 7.854 Rad
Conclusion
Converting between radians and degrees is an essential skill for trigonometry, physics, and chemistry. And fortunately it’s not that hard for you to acquire this skill. Following all the tips and tricks you’ve seen here today will make sure of that.