Degrees to Radians

How to Convert Degrees to Radians

Degrees and radians are two different units that are used for the measurement of the angles. The conversion of degrees to radians is considered while measuring the angles in Geometry. The measure of the angle is generally denoted by degrees, having the symbol °. An angle can be determined by two different kinds of units, that are, degrees and radians. You can convert one form of the representation of any mathematical angle to the other by using simple formulas. A degree also has its sub-parts that are minutes and seconds. This conversion plays a major part in the trigonometry applications. In this article, we will learn about how to convert degrees to radians, degrees to radians formula, and look at some solved examples based on how to convert degrees to radians formula. Let us first look at the degree to radian conversion.


The Degree to Radian Conversion

The value of 180° equals to \[\pi\] radians. For converting any given angle from the measure of its degrees to the radians, you need to multiply the value by \[\frac{\pi}{180}\].

Angle in radians = angle in degrees x \[\frac{\pi}{180}\]

The value of \[\pi\] = 22/7 or 3.14


Degrees to Radians Formula

We already learned about how to change degrees to radians for any given angle. Let us learn about how to convert degrees to radians formula. The formula to change the degree to radian is given as follows:

Degree x \[\frac{\pi}{180}\] = radians


How to Convert Degrees to Radians

Let us now look at a step by step procedure for converting degrees to radians.

  1. Jot down the degrees that you want to convert into radians. Consider the following examples: 

Example 1: 120°

Example 2: 30°

Example 3: 225°


  1. Then multiply the degrees by π/180. 

Example 1: 120 x π/180

Example 2: 30 x π/180

Example 3: 225 x π/180


  1. Then simply carry out the multiplication by multiplying the degrees by π/180. Think of it as if you are multiplying two fractions. The first fraction consists of the degrees in the numerator and 1 as the denominator, and the second fraction consists of π in the numerator and has180 in the denominator.

Example 1: 120 x π/180 

= 120π/180

Example 2: 30 x π/180 

= 30π/180

Example 3: 225 x π/180 

= 225π/180


  1. The last step is to simplify. Now, you have to put each fraction in its lowest terms to get the final answer. Find the largest number which can evenly divide to the numerator and the denominator of each fraction and use it for simplifying each fraction. 

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

This is a very simple method and you can easily convert degrees to radians with this simple procedure. Let us look at how to convert angles to radians in detail.


How to Convert Angles to Radians

You learned about how to change degrees to radians. Now let us learn about how to convert the angle to radians. 

The angle that is made when the radius is wrapped around the circle is given as follows:

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1 radian is approximately 57.2958°.

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Consider a half-circle which has radians, which equals to 180°.

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Hence, 1 radian = \[\frac{180^{0}}{\pi}\]= 57.2958°

If you want to convert a degree or an angle to radians, simply multiply the angle by and then divide it by 180.

Take a look at the table below of the angles and their conversion to radians.

Degrees 

Radians 

Approximate radians

30°

𝜋/6

0.524

45°

𝜋/4

0.785

60°

𝜋/3

1.047

90°

𝜋/2

1.571

180°

𝜋

3.142

270°

3𝜋/2

4.712

360°

2𝜋

6.283


Solved Examples

Let us now look at some of the solved examples for your better understanding of how do you convert degrees to radians and radians to degrees.

Example 1

Convert 120° to radians.

Solution:

To convert 120° to radians, consider the formula

Angle in radians = angle in degrees x  (𝜋/180)


Therefore, 120° x (𝜋/180)

= (2𝜋/3) radians ≈ 2.09 radians


Example 2

Convert 1.4 radians to degrees.

Solution:

To convert radians to degrees, consider the following formula:

Degree =  radians  x 180/𝜋

= 1.4 180 = 252

= 1.4 x 180/𝜋 = 252/𝜋

≈  80.2° 

FAQ (Frequently Asked Questions)

1. How To Convert 1 Degree To Radians?

There is a simple formula for converting radians to degrees. 

1π radian = 180°.

Therefore you can convert one unit of measure to the other easily.

One degree is equal to 0.01745329252 radians.

Therefore,

1° = π/180° 

= 0.005555556π

= 0.01745329252 rad

2. What is a Radian?

The measure of ant given angle can be determined by the amount of the rotation that takes place from the initial side to the terminal side.  One way in which you can measure angles is in radians.  To define a radian, use the central angle of a circle. Consider an angle whose vertex is the centre of the circle.  One radian here refers to the measure of the central angle which intercepts the arc s that is equal in the length to the radius r of the given circle.

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As you know that the circumference of a circle is given by 2πr one revolution around the given circle having a radius r corresponds to the angle of 2π radians since 

s/r = (2πr/r)

= 2π