Degrees to Radians: Overview, Questions, Preparation

An angle can be measured in both degrees and radians. When an object finishes one revolution from point A to point A in an anticlockwise direction, it can either be denoted by 360 degrees or 2π when measured in radians. 

Formula to Convert Degrees to Radians and vice-versa:

• Degree to radian conversion 

One radian is equal to 180 degrees. Therefore, to convert the measurement of an angle from degrees to radians, we will multiply it by π/180.

Therefore, the degree to radian formula is given by Degree x π/180.

The above image shows a pictorial representation of 1, -1, 1 ½ and -1 ½ radians. 

• Radian to degree conversion 

Conversely, we can use the following formula while converting the measurement of an angle from radians to degrees:

Radian to degree formula = Radians x 180/π.

To convert a negative degree to radians, you need to follow the same formulas as when the degree is positive. However, instead of multiplying the positive degree here, we will multiply π/180 by the degree’s negative value. 

Therefore, the formula to convert a negative degree to radians can be given as below:

Degree to Radians (Negative) = π/180 x (-degrees)

Conversion Chart: Degrees to Radians 

The table below shows the values of angles in degrees and their corresponding values in radians:

The degree to radian conversion and radian to degree conversion is covered in the topic of Trigonometric functions in Class X. It carries a weightage of 2 to 4 marks as it is a simple conversion.

1. Convert 150 degrees to radians.

Sol: The formula to convert degrees to radians is given below:

Angle in radians = Degrees x π/180 = 150 x π/180 = 5 x π/6

Now, as per the degree to radian conversion chart, π/6 = 0.524

Therefore, angle in radians = 5 x π/6 = 5 x 0.524 = 2.62 radians. 

2. Convert 480 degrees into radians. 

Sol: As per the formula, Angle in radians = π/180 x degrees = π/180 x 480 

= (π/3) x 8 

As per the degree to radian conversion chart, π/3 = 1. 047

Therefore, Angle in radians = 1.047 x 8 = 8.376 radians.

3. Convert 10 radians into degrees. 

Sol: As per the formula to convert radians to degrees, 

Angle in degrees = 180/π x radians = 180/π x 10 

Now, the value of π is 3.142 approximately as per the degree to radian conversion table. If we substitute this value for π in the given example, we get

Angle in degrees = 180/3.142 x 10 = 57.29 x 10 = 572.9 degrees approximately. 

Q: How much is the value of one degree in radians?

Q: How much is the value of one radian in degrees?

Q: Where is the use of radians for angle measurement more convenient in mathematics?

Q: How is a circle represented in radians?

Q: Where did the term radian first appear?