Radians
Radians are a unit that measure angles. A full rotation in radians is equal to (tau) radians, where the constant is approximately equal to . Radians are the preferred unit for measuring angles, as opposed to degrees, since they lead to “more succinct and elegant formulas throughout mathematics"^{[1]}.
A radian angle is equivalent to the ratio of the arclength over the radius of the circle. By design, the unit is “dimensionless” or, in plain language, the length of the radius is arbitrary; one radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle. The symbol (equivalent) is used to indicate that negative angles or angles more than a full rotation are mapped to the equivalent angle between and a full rotation.
This “radiusinvariant” property plays an important role for why radians are the preferred unit for measuring angles. See radians vs. degrees.
The radian system is used in many places in mathematics: measuring angles, numerous geometric and advanced math formulas via the circle constant, and as the unit of choice for the trigonometric functions.
The circle contanstant (tau) makes talking about and using the radians angle system much more convenient. The circle constant is approximately equal to ^{[2]} as illustrated by figure 3.
Without the circle constant, angles measured in radians can be confusing. Take for example, the figures figure 4 and figure 5 showing measured angles of 1 radians and 2 radians. Both figures draw tic marks at the numeric values of radians, which conceptually demonstrates what a radian is. However, as we will later see it is much more convenient to use a constant.
Angles measured using radians are expressed as a fraction of “one turn” where one turn is equal to the circle constant (tau). By convention, angles in the XY
coordinate plane are measured from the positive direction where the counterclockwise rotation is positive.
This is where the circle constant plays a crucial role in making the radians system complete; angles can easily be expressed as fractions of the whole. As figure 6 and figure 7 illustrate, the circle can be divided into equal parts and using the circle constant constant leads to elegant and succinct notation.
As a final note on measuring angles, here are some “special angles” that show up in numerous places. The figure also illustrates why circle constant is also the best method for converting between radians and degrees^{[3]}
Shown below are some special angles measured using radians. Note, the symbol (theta) is often used to represent a measured angle.
Radians appear in numerous geometric formulas and advanced formulas in mathematics via the circle constant or or alternatively through the constant . Shown below are some common geometric formulas. Note, very likely you have seen the constant used insted of This website primarily uses instead of in equations and formulas as it is the better constant^{[4]}. The relation can be used to translate between the two constants.
The area of a circle is give by onehalf multiplied by τ (tau) mutliplied by the radius of the circle squared.
The circumference of a circle is given the constant τ (tau) multpilied by the radius of the circle, where τ = 2π.
The volume of a sphere is given by twothirds multiplied by the circle constant τ (tau) multiplied by the radius cubed.
The circle constant also appears in some suprising places such as the definition of the normal distribution. When is present, there usually is some connection to circles or a circular way of thinking.
The trigonometric functions and the unit circle are often where radians are introduced. Some programming languages and spreadsheet applications only offer the trigonometric functions implemented using the radian system. This is discussed in more detail on the individual function pages, but shown below are two figures which graph the two functions using the radian system.
There are two systems for measuring degrees in math: degrees and radians. The short answer for why radians are preferred is that the radian system leads to more succinct and elegant formulas throughout mathematics ^{[4]}. For example, the derivative of the sine function is only true when the angle is expressed in radians.
See the long answer for why radians are preferred on this page.
To convert an angle from radians to degrees multiply by the ratio of over radians ^{[3]}.
The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.
The greek letter π (pi) is a geometric constant approximately equal to 3.1456. The numeric value is equal to the length of any circle's circumference divided by its diameter.
An angle is defined as the amount of rotation between two rays. Angles are measured using degrees and radians. A full rotation in degrees is 360°. A full rotation in radians is approximately 6.283 radians or τ (tau) radians.
This page compares and contrasts the two systems of measuring angles in math: radians and degrees, and explains why radians is the preferred unit of measure for angles.
Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.
There are six trigonometric functions that relate to the geometry of the righttriangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the function forms a "bellcurve" which is symmetric around the mean and whose shape is described by the standard deviation.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.
Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Radians Versus DegreesWumbo (internal)

Approximate the Circle ConstantWumbo (internal)

Convert Angle from Radians to DegreesWumbo (internal)

No, really, pi is wrong: The Tau ManifestoMichael Hartl