Degree and Radian Measure Formula

Radian and Degree Measure

The measure of an angle is identified by the amount of rotation from the original side to the end side. In radians, one complete counterclockwise rotation is 2π. Whereas, in degrees, one complete counterclockwise rotation is 360°. Thus, the degree measure and radian measure are linked by the equations

360°= 2π radians


180°= π radians

From the latter, we get the equation 1° = π180 radians.

This results in the rule to convert degree measure to radian measure. In order of conversion from degrees to radians, multiply the degrees by π180° radians.


How to Convert Degrees to Radians?

The value of 180° is equivalent to π radians. In order to convert any given angle from the measure of degrees to radians, the value needs to be multiplied by π/180. A standard formula for converting from degrees to radians is to simply multiply the number of degrees by π/180

Thus, Angle in Radian = Angle in degree × π / 180° 

Radian Measure Formula

In any circle of radius r, the ratio of the arc length ℓ to the circumference is equivalent to the ratio of the angle θ subtended by the arc at the midpoint and the angle in one rotation. Therefore, radian formula to measure the angles in radians, ℓ2πr=θ2π⟹ ℓ=rθ.

Degree Calculation Formula

The measure of the angle is considered in degrees (°). Radian is often considered while computing the angles of trigonometric functions or periodic functions. Radians are always denoted in terms of pi (π), where the value of pi is equivalent to 22/7 or 3.14.

A degree consists of sub-parts also, stated as minutes and seconds. This conversion is the crucial component of Trigonometry applications. Thus, the Degree formula is: - Radians × 180/π = Degrees

Solved Examples on How to Find Radian Measure From Degree


Convert 135° into radian measure


Given = Angle of 135 degrees

135∘× π/180∘

=3π/4 radians

≈2.35 radians


Convert 210 degrees to radians.

Solution: Given = Angle of 210 degrees

Angle in radian = Angle in degree x (π/180)

= 210 x (π/180)

= 7π/6


Hence, 210 degrees is equal to 7π/6 (3.67) in radian.

FAQs (Frequently Asked Questions)

1. What is Meant By a Radian?

Answer: In mathematics, the radian is said to be the standard unit of angular measure. An angle’s measurement in radians is numerically equivalent to the length of a concurrent arc of a unit circle. The link or the relationship between the arc length and radius of a circle describes the radian of a circle. Degree formula and radian formula are used to convert, degree to radian or radian to degree.

Radian= Arc Length / Radius Length

Radian=Degree × π / 180

2. What is the Deal of a Circle With Radians Anyway?

Answer: Most of you must be thinking of a circle with respect to degrees: 360° is a complete circle. 180° makes for a half the circle etc... Well, radian measure or radian measure formula is just a distinctive way of talking about the circle or measuring the angle of a circle. Radian measure is just another unit of measure.

Similar to the way we can measure a cricket ground in yards or feet -- we can measure a circle in degrees (like the olden days) or in radians (welcome to the big federation!)

Radian also has a close connection to the radius of a circle.