# Convert Angles from Degrees to Radians - Trigonometry Calculator

An online calculator to convert angles from degrees to radians.

## Convert Degrees to Radians

Degrees and radians are units of measures of angles in trigonometry and geometry; and in many situations a conversion from degrees to radians in needed.

A full rotation correspond to an angle of 360° or 2? radians as shown in the diagram below.
We can say that to each degree corresponds 2?/360 radians.

Hence the degrees to radians formula

angle in radians = angle in degrees × 2 ? / 360

which may be simplified, by dividing numerator and denominator by 2, to

angle in radians = angle in degrees × ? / 180

Example 1: Convert x = 120° to radians

x = 120 × ? / 180 = 2 ? / 3 = 2.094 radians

Example 2: Convert x = - 20° to radians

x = - 20 × ? / 180 = - ? / 9 = - 0.349 radians

Example 3: Convert x = 34° 30' 45" (degrees, minutes, seconds) to radians

We first convert x to a decimal form as follows

x = 34 + 30/60 + 45/3600 = 34.5125 degrees

then into radians

x = 34.5125 × ? / 180 = 0.692 radians

## Applications of Conversion from Degrees to Radians

Example 4: Calculate the arc length corresponding to a central angle ? = 55° on a circle of radius r = 10 cm.

Solution

The formula for the arc length is

Arc length = ? × r , where ? is in radians.

We first convert ? in radians

? = 55 × ? / 180

Arc length = (55 × ? / 180) × 10 = 9.6 cm

Note: There are many formulas in Mathematics and Physics involving angles in radians, hence the need for conversion in order to use these formulas.

Example 5: Compare the size of angle ? = 23° and angle ? = 0.43 radians.

Solution

We need to convert the measure of angle ? to radians and then compare them.

? = 23× ? / 180 = 0.40 radians.

It is now easy to conclude that ? is larger that ?

Example 6:

a) In calculus, it is preferable to use radians (similar to no unit) in trigonometric functions otherwise expressions for the derivatives of sine, cosine, ... and many other concepts will involve constants that will make things difficult.

let y = sin(x)

If x is in radians , dy/dx = cos(x)

If x is in degrees , dy/dx = (?/180) cos(x)

b) The expansion of sin(x) as a series

sin(x) = x + x^{ 3}/3! + x^{ 5}/5! + ...

is valid for x in radians.

## How to Use the Calculator to Convert Degrees to radians?

There are two calculators where the inputs differ.

1 - Enter the size of the angle in __degrees as a decimal number__ (such as 30.0, 25.8,...) and press "enter". The answer is the size of the same angle in radians. There are two possible outputs. The first in decimal form and the second as a fraction of ?. For example for an input of 30°, the decimal output is 0.5236 and the fraction of ? output is 1/6?.

2 - Enter the size of the angle in __degrees, minutes and seconds__ (such as 10° 12' 34") and press "enter". The answer is the size of the same angle in radians in decimal form.
You may also change the number of decimal places for the outputs. The input must be a positive integer.
## More References and Links

Convert Angles From Radians to Degrees

Angles In Trigonometry

Convert Angles from Decimal to Degrees, Minutes and Seconds

Convert Angles from Degrees, Minutes and Seconds to Decimal

Math Calculators and Solvers.