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Principles of Physics

Radians and degrees

by Angel G. Fuentes, MS

R

Fields of Study

Classical Mechanics; Electronics; Harmonics

Summary

From the development of astronomy by ancient civilizations to the internal combustion engine, rotations about a center have been an integral part of the development of scientific theories and advances in engineering. Though its origins are not known exactly, the degree as a unit of measurement was originally developed by Babylonians and Persians. The concept of the radian was developed in the early eighteenth century by British mathematician Roger Cotes. This article explains these basic units of angular separation.

Principal Terms

  • angle: the separation along a curved path between two rays originating at the same point.

  • arclength: the length of an arc defined by an angle and two radii.

  • circumference: the distance around a circle defined by a radius.

  • cosine: in a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.

  • pi: the ratio of the circumference of a circle to its diameter.

  • radius: distance from the center of a circle to any point along the circle.

  • sine: in a right triangle, the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.

  • tangent: in a right triangle, the ratio of the length of the side opposite an acute angle to the length of the side adjacent to the same angle.

Origins of Degrees and Radians

For centuries, scientists and mathematicians used the unit of degrees (°) to measure the size of an angle. The origin of the degree is not exactly known. One theory is that it originated in ancient astronomy. Previous civilizations started to notice that the positions of the stars were the same every 360 days—very close to a year. To represent a year, Babylonian and Persian mathematicians divided the circumference of a circle into 360 spaces or units. This unit became known in ancient Greece as the moira, which was translated into Arabic as daraja, meaning a step on a scale or ladder. In Latin, this became de gradus, and in English, degree. With the development of trigonometry, the mathematics of angles, mathematicians and scientists found they needed a way to represent angular measurements as numeric values without arbitrarily determined units such as degrees.

A circle divided into twelve sections with angle measurements in degrees and radians for each section. Each section is 30 degrees or 1/6th of the value for pi. Although both of these units are used to measure angles, they cannot both be used in formulas for other calculations. The length of an arc (d ) can be calculated by multiplying the radius by the angle in radians, but not in degrees.

POP_RadiansDegrees_illustration.tif

Sample Problem

Consider two angles, one 125° and the other 5π/7 rad. What are the values of the angles in radians and degrees, respectively, given that π has an approximate value of 3.14?

Answer:

To calculate the first angle in radians, multiply 125° by π/180:

POP_RadiansandDegrees_equation5.tif

The first angle measures approximately 2.18 radians.

To calculate the second angle in degrees, multiply 5π/7 rad by 180/π:

POP_RadiansandDegrees_equation6.tif

The second angle measures approximately 128.57 degrees.

In 1714, British mathematician Roger Cotes (1682–1716) published a paper titled “Logometria.” In it, he developed the idea of an angular unit derived from the ratio of an arclength along the circumference of a circle to the radius of the circle. He defined this unit, now known as the radian, as the angle corresponding to an arclength that is equal to the radius of the circle. This measurement of angular separation is approximately equal to 57.3 degrees. The unit of radians uses no arbitrary definitions as to what it means. It can therefore easily be used in mathematical applications ranging from trigonometry to calculus. In trigonometry, the basic functions of the sine, cosine, and tangent of an angle are defined as the ratios between the lengths of two sides of a right triangle. Since the radian is a ratio of two lengths, it has no units and can be used in trigonometry. In calculus, the need for a unit-less quantity was clear when trying to calculate simple harmonic motions of molecules and other simple oscillators.

Calculating Radians

The circumference of a circle is equal to 2πr, where r is the radius of the circle and π is pi, a mathematical constant approximately equal to 3.14. Because a radian corresponds to the central angle of an arclength equal to one radius, there are 2π radians in a circle. A full circle is 360 degrees around. For a circle with a radius equal to one, therefore, one radian is equal to 360/2π degrees, or 180/π degrees. Conversely, one degree is equal to π/180 radians. Multiply the measurement of an angle in degrees (θd) by π/180 to convert the angle to radians (θr):

POP_RadiansandDegrees_equation1.tif

For example, if an angle measures 120°, that same angle in radians (rad) is

POP_RadiansandDegrees_equation2.tif

To convert radians to degrees, simply multiply by 180/π:

POP_RadiansandDegrees_equation3.tif

For example, if an angle measures π/2 rad, that angle in degrees is

POP_RadiansandDegrees_equation4.tif

Applications of Radians and Degrees

The radian and degree both remain useful units of measurement. The degree is still used in navigation and astronomy. It is also used by the general public to explain rotations and revolutions. On the other hand, the radian holds a prominent role in scientific and mathematic developments. It is the International System of Units (SI) derived unit of angular measurement. Without the use of radians, scientists would not be able to properly calibrate electric generators and the angular frequencies at which they spin.

Citation Types

MLA 9th
Fuentes, Angel G. "Radians And Degrees." Principles of Physics, edited by Donald R. Franceschetti, Salem Press, 2016. Salem Online, online.salempress.com/articleDetails.do?articleName=POP_0101.
APA 7th
Fuentes, A. G. (2016). Radians and degrees. In D. R. Franceschetti (Ed.), Principles of Physics. Salem Press. online.salempress.com.
CMOS 17th
Fuentes, Angel G. "Radians And Degrees." Edited by Donald R. Franceschetti. Principles of Physics. Hackensack: Salem Press, 2016. Accessed September 20, 2026. online.salempress.com.