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Maths > Circles > Radian – Definition, Formula, Examples

Radian – Definition, Formula, Examples

What is a Radian?

Firstly, the radian is a unit for measuring the angles that we mainly use in the trigonometry. We use it instead of the degrees. Whereas a full sphere or a circle is of 360 degrees, a full sphere or a circle is just above six radians.


Definition of Radian

The radian is an SI unit that helps in the measurement of the angles. Moreover, it is also the standard unit for angular measurement that we use in various areas of mathematics. The unit circle’s length of an arc is number wise equal to the measurement in radians of the angle that it subtends.

A single radian is just below 57.3 degrees that expands at OEIS: A072097. However, the unit was formerly an SI unit by the supplement. Since 1995 the radian is an SI derived unit.

Separately, the SI unit of the solid angle measurement is steradian. Moreover, the radian is most usually represents itself by the symbol rad. Its alternative symbol is C. The superscript letter c, the letter r, or a superscript R.

Whereas these symbols are infrequent and we can use them as they can be easily mistaken for a degree symbol (°), or radius (r). For example, a value of 1.2 radians is capable of writing as 1.2 rad, 1.2 r, 1.2rad, 1.2R or 1.2C.

As a ratio of the 2 lengths, a radian is a pure number that requires no unit symbol. In mechanical writing, the symbol rad is almost omitted every time. However, when quantifying an angle in the absence of some angle, the radians are set on an assumption. When the degrees are meant then the symbol ° is used.

Formula of Radian

Firstly, One radian = 180/PI degrees and one degree = PI/180 radians. Therefore, for converting a specific number of degrees in radians, multiply the number of degrees by PI/180 (for example, 90 degrees = 90 x PI/180 radians = PI/2). To convert a specific number of radians into degrees, multiply the number of radians by 180/PI.

Example of Radian

The full angle is 2 pie radians and therefore it is 360 degree.
A right angle is pie/2 radians.
A straight angle is pie radians.

Advantages of Measuring in Radians

In calculus and most of the other branches of the mathematics beyond the practical geometry. The angles are measurable in radians. Moreover, it happens because radians consist of a mathematical naturalness that results in a more elegant formulation of a number of important outcomes.

Most notably, results in an analysis that involves trigonometry also are expressed in radians. For example, the use of radians results in the most simple limit formula.

Which is the basis of many other identities in mathematics that also includes:

Dimensional Analysis

Although the radian is a unit for measurement. It is a dimensionless quantity. This is visible from the definition given earlier above. The angle subtended at the center of the circle, measured in radians. This angle is equal to the ratio of the length of the enclosed arc to the length of the radius of the circle

Therefore the units of measurement cancel, this ratio is without any dimension. Although polar and the spherical coordinates uses radians to describe coordinates in 2 and sometimes 3 dimensions. The unit is derivable from the radius coordinate, so the angle measure is still without any dimension.

Use in Physics

The radian is useful for physics widely when the requirement of the angular measurement arises. For example, angular velocity is typically measurable in radians per second. (Rad/s). One revolution per second gets equals to 2 pie radians per second.

Solved Question For You

Ques. Convert 5/12 π radians into degree measure:

  1. 75°
  2. 15°
  3. 180°
  4. 432°

Ans. A

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