Radians in Mathematics Definition and Meaning

Radians and degrees are two different ways to measure an angle. To measure the angle in degrees, we need to consider the one complete circle of 360º. Further, by dividing the one complete circle proportionally into 360 parts, each of these parts of the circle is 1º, and for any integer 1 ≤ d ≤ 360º, the measure of the d combined parts obtains dº. 

Another system of measurement is by radians, in which angle is measured by measuring the arc length of the  unit circle, 

Here, we will learn the terms radians and degree in detail.

What are Radians?

The radian (represented by the symbol rad) is the SI unit used to measure angles, and is the angular measure used in the different areas of Mathematics. The unit radian was earlier known as the SI supplementary unit, but now it is considered as an SI derived unit. One radian is defined as the angle subtended at the centre of the circle by an arc whose length is equal to the radius of the circle. 

Radians Formula

The angle formed by the half circle is 180º which is equivalent to π radians.

Accordingly, the angle formed by one complete counterclockwise revolution is equivalent to 2π radians.

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Hence, the radians formula in terms of π 

1 Radians = 360^o/2π

How to Measure Angles in Radians and Degree?

Consider a unit circle given below centred at the origin and let represents an angle formed with the positive x - axis. Then the measurement of an angle in radians is the arc length of the circle turned out by the angle is 0.

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To understand how to calculate this, let us discuss a few examples. If the given angle measures 0 in degree, then the arc length of the circle turned out by an angle is 0. Similarly, If the given angle measures 360 degrees, then the arc length of the circle turned out by an angle is the circumference of a circle with radius equal to 1 or 2π. 1 = 2π . For any other angle measurement in degrees, we need to compute the fraction of the arc length turned out by converting from degrees to radians, as shown in the example given below.

Example

Find the Value of the Right Angle in Radians.

Solution:

As the value of right angle is 90º in degrees

Accordingly, the value of 90º in radians  = 90ºx π/180 = π/2

Therefore, the value of right angle or 90º in radians is equal to π.

Radius and Degrees Formula

Radius and degrees formulas given below helps us to find the angles given in degrees to radians, and vice versa. 

To convert an angle given in degrees to radians , simply multiply the angle by π/180

To convert an angle given in radians  to degrees , simply multiply the angle by 180/π.

Another simple formula to convert degrees to radians is :

\[\frac{D}{90}=\frac{R}{\frac{\pi }{2}}\]

The above formula to convert degrees to radians is true because the number of radians in 90º is equal to  π/2 radians. Hence, we can convert the angle given in degree to radians using the above formula also.

Degrees and Radians Differences

• A degree is a unit of measurement that is used to measure spheres, circles and angles whereas radians is a unit of measurement that is only used to measure angles.

• A degree is a unit of measurement based on the amount of rotation or turn while a radian is a unit of measurement based on the arc length obtained by each angle.

• Degrees are further classified into minutes and arc seconds whereas radians have no subdivision but use decimals for smaller and fractional angles. 

• A degree is a unit used to measure angle which equals 1/360 of one complete circle whereas radians is the angle subtended by a circular arc which has a similar length as its radius. Radians states that the circle follows 360 or 2π radians.

• Degrees are most commonly used in simple Mathematics whereas radians are used in advanced Mathematics.

How to Use a Radian Calculator?

Following are the steps that should be followed to use a radian calculator.

Step 1: Enter the angle in radians in the input column.

Step 2: Click on convert to degrees button

Step 3: As you click on the button, the conversion value from radian to degree will be displayed in the output column.

How to Use a Degree to a Radian Calculator?

Following are the steps that should be followed to use a degree to radians calculator.

Step 1: Enter the angle in degrees in the input column 

Step 2: Click on convert to radians button

Step 3: As you click on the button, the conversion value from degree to radians will be displayed in the output column.

Degrees to Radians Conversion Values Table

Fun Facts

• In one whole circle, there are 2π radians, which implies that half a circle (180 degrees) is equal to π radians.

• 1 Radians = 180/π or 57.2958 degrees.

Radians and Degrees Examples With Solutions

1. Convert the Following From Radians to Degrees.

1. π radians

2. π/2 radians

Solution:

To convert the given values in radians to degrees, we use the following radians to degree formula:

Radians = 180/π (Angle in Degrees)

1. π radians = π x 180/π = 180º

2. π/2 radians = π/2 x 180/π = 90º

2. Convert the Following From Radians to Degrees

1. 60º

2. 180º

Solution:

To convert the given values in degrees to radians, we use the following radians to degree formula:

Degrees = π/180 (Angle in Radians)

1. 60^o x π/180 =π/3

2. 180^o x π/180 = π