Formula, Definition | Radians And Degrees - Cuemath

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 1.5c.

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, 6c, 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 1c 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:

• Trigonometric Ratios in Radians
• Degrees to Radians Calculator
• Radians to Degrees Calculator