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Radian
Radian is a unit used to measure angles. We have two units to measure the angles: degree and radian. Up to this stage, you might have been using degrees to measure the sizes of angles. However, for a variety of reasons, angle measures in advanced mathematics are frequently described using a unit system different from the degree system. This system is known as the radian system. Did you know that radian was the SI supplementary unit for measuring angles before 1995? It was later changed to a derived unit.
Come, let us learn in detail about the radian formula, the arc length formula, and how to convert an angle from radians to degrees and from degrees to radians.
1.  What is Radian? 
2.  Radian Formula 
3.  Conversion Between Radians and Degrees 
4.  Differences Between Radians and Degrees 
5.  FAQs on Radian 
What is Radian?
The radian is an S.I. unit that is used to measure angles and one radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. A single radian which is shown just below is approximately equal to 57.296 degrees. We use radians in place of degrees when we want to calculate the angle in terms of radius. As '°' is used to represent degree, rad or ^{c} is used to represent radians. For example, 1.5 radians is written as 1.5 rad or 1.5^{c}.
Radian Definition
"Radian" is a unit of measurement of an angle. Here are few facts about "radian"
 Radian is denoted by "rad" or using the symbol "c" in the exponent.
 If an angle is written without any units, then it means that it is in radians.
 Some examples of angles in radians are, 2 rad, π/2, π/3, 6^{c}, etc.
Radian Uses
 Angles are most or generally measured in radians in calculus and in most other branches of mathematics.
 Radians are widely used in physics also. They are preferred over degrees when angular measurements are done in physics.
Radian Formula
We have already learned that 1 radian is equal to the angle made by the arc of a circle whose length is same as the radius of the circle. Thus, the angle subtended by an arc in radians of a circle is defined as the ratio of the arc length to the radius of the circle.
If we consider the arc to be the total circumference of the circle, then arc length = 2πr. Also, we know that the angle subtended at the center of the circle by its circumference is 360°. Then by the above formula,
Angle subtended = (arc length)/(radius)
360° = (2πr)/r
360° = 2π
Thus, the formula of radians is 2π = 360°.
Conversion Between Radians and Degrees
An angle can be converted from "radians to degrees" and from "degrees to radians" according to necessity. We use the radian formula (from the previous section), 2π = 360° for doing these conversions. We can see how to do the conversions between the radians and degrees in the figure below.
Converting Radians to Degrees
The radian formula can be written as,
2π Radians = 360°
From this, 1 Radian = 360°/2π (or)
1 Radian = 180°/π.
Thus, to convert radians to degrees, we multiply the angle by 180°/π.
Examples of Converting Radians to Degrees:
 π/2 = π/2 × 180°/π = 90°
 π/4 = π/4 × 180°/π = 45°
 7π/6 = 7π/6 × 180°/π = 210°
 2 rad = 2 × 180°/π ≈ 114.59°
If we observe the first three examples where the angle is in terms of π, π is getting canceled while converting it into degrees. So to convert an angle in radians that is in terms of π into degrees, just replace π with 180°. This is a trick to convert radians into degrees. Here you can see the first three examples using the trick.
 π/2 = 180°/2 = 90°
 π/4 = 180°/4 = 45°
 7π/6 = 7(180°)/6 = 210°
Converting Degrees to Radians
From the radian formula,
2π Radians = 360°
From this, 1° = (2π Radians)/360°
1° = (π/180) radians
Thus, to convert degrees to radians, we multiply the angle by π/180 radians.
Examples of Converting Degrees to Radians:
 90° = 90 × π/180 = π/2
 180° = 180 × π/180 = π
 210° = 210 × π/180 = 7π/6
Radians and Degrees Table
Here is a table with some standard angles in degrees and the corresponding angles in radians. This table is helpful to know the equivalent angles of radians (or degrees).
Degree  Radian 

30°  π/6 
45°  π/4 
60°  π/3 
90°  π/2 
180°  π 
270°  3π/2 
360°  2π 
Differences Between Radians and Degrees
Radians and degrees both are measurements of angles only. Here are a few differences between radians and degrees.
Radian  Degree 

The angle subtended by an arc of length 'r' of a circle whose radius is 'r' is known as 1 radian.  1/360 ^{th} part of a complete angle is called a degree. 
1 radian is denoted by 1^{c} or sometimes simply 1.  1 degree is denoted by 1°. 
To convert an angle from degrees to radians, multiply it by π/180.  To convert an angle from radians to degrees, multiply it by 180/π. 
Important Notes on Radian:
 We can convert an angle from degrees to radians by multiplying it with π/180.
 We can convert an angle from radians to degrees by multiplying it with 180/π.
 Arc length = radius × angle subtended at the center.
While applying this formula, the angle (if given in degrees) should first be converted to radian.
☛ Related Topics:
Examples on Radians

Example 1: Convert the following angles into radians. a) 120° b) 150°.
Solution:
To convert the given angles into radians, we multiply each of them by π/180.
a) 120° = 120 × π/180 = 2π/3
b) 150° = 150 × π/180 = 5π/6
Answer: a) 120° = 2π/3 b) 150° = 5π/6.

Example 2: What is the arc length of a circle whose radius is 6 inches and the angle subtended by the arc is 1.5 radians?
Solution:
The radius of the circle = 6 inches.
The angle subtended by the arc = 1.5 radians.
We know that the arc length is the product of the radius and the angle subtended by the arc at the center of the circle.
So arc length = (6)(1.5) = 9 inches.
Answer: Arc length = 9 inches.

Example 3: A pendulum of length 18 inches oscillates at an angle of 42 degrees. Find the length of the arc that it covers. Express the answer in terms of π.
Solution:
The radius of the circle = 18 inches.
The angle subtended by the arc = 42° = 42π/180 radians.
The arc length = (18)(42π/180) = 4.2π inches.
Answer: The length of the arc that is covered by the pendulum = 4.2π inches.
FAQs on Radian
What is a Radian?
Radian is the SI unit of measuring angles which is based on the arc length and the radius. 1 radian is equal to the angle subtended by an arc at the center of the circle whose length is equal to the radius of the circle.
What is the Radians to Degrees Formula?
To convert radians to degrees, we multiply the angle by 180/π. For example, 3π/2 = 3π/2 × 180/π = 270°. Thus, the degree equivalent angle of 3π/2 is 270°.
What is the Degrees to Radians Formula?
To convert degrees to radians, we multiply the angle by π/180. For example, 270° = 270 × π/180 = 3π/2 . Thus, the radian equivalent angle of 270° is 3π/2.
How many Radians are there in a Complete Circle?
In a full circle, the angle is 360° and by radian formula, 360° = 2π. Thus, there are 2π radians in a complete circle.
What is Arc Length Formula in Terms of Radians?
The length of an arc of a circle is the product of its radius and the angle subtended by the arc in radians. i.e., arc length = radius × angle in radians. If the angle is given in degrees, we first have to convert this angle into radians and then apply the arc length formula.
What is the Radian Formula in Terms of Arc Length?
To find the angle subtended by an arc in radians, we just divide its length by the radius of the circle. i.e., θ (in radians) = (arc length) / radius.
What is 1 Radian Equal to?
We know that 2π radians = 360°. Dividing this equation by 2 on both sides, π radians = 180°. Dividing this equation by π on both sides, 1 radian = 180°/π. Thus, 1 radian = 57.296°. This can be used to convert any angle into radians. For example, 2 radians in degrees = 2 × 57.296 = 114.592°.
What is the Measure of 0 Radian in Degrees?
For converting 0 radians into degrees, we multiply it by 180/π. Thus, 0 radians = 0 × 180/π = 0°. i.e., 0 radians is nothing but 0 degrees.
What is 360 Degrees in Radians?
To convert 360 degrees to radians, we multiply the angle by π/180. 360° = 360 × π/180 = 2π . Thus, the radian equivalent angle of 360° is 2π.
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