How to Convert Degrees to Radians? Formula, Equation, Conversion


When measuring angles in geometry, degrees to radians is a way of converting the unit of measurement. Angles are also measured in different units according to their field of application, just as every quantity has a unit of measurement. The units of measurement for angles are degrees and radians. Because most angles are measured in radians instead of degrees, it is essential to know how to convert degrees to radians.

Radian measures can be applied to sectors of a circle, arc lengths, and angular velocity, among others. If an object is moving in circular paths or parts of a circular path, one should use radians. Angles in degrees can be calculated in the problem statement, but they should always be converted to radians before being used in calculations. During this article, we will discuss degrees to radians formula and conversion table with examples.

How to Convert Degrees to Radians?

Define Radian

In geometry, radians are another unit for measuring angles. At the center of a circle, a radian is an angle formed by an arc whose length equals the circle’s radius r.

how convert degrees radians formula equation conversion example
how convert degrees radians formula equation conversion example

Rotations counterclockwise are expressed as 2 radians. Straight angles are also expressed as radians, and right angles are expressed as π/2 radians. When we compare the angle measurement of a complete rotation in radian and degrees to this, we see the following:

  • 180 degree = π radian
  • 360 degree = 2π radian

Invention of Radian

Roger Cotes is generally credited with developing radian measurement in 1714, as opposed to the degree of an angle. The radian was described in everything except its name, and its naturalness as a unit of angular measurement was recognized.

Degrees to Radians Conversation Table

In the following table, we have provided the radian values for corresponding degrees angles. In this table, degrees are converted to radian angles from 0 to 360 degrees.

Degree0153045607590105120135150180240270360
Radian0π /12π / 6π /4π / 35π/12π / 27π/122π / 33π / 45π/6  π4π/3  3π / 2

Degrees to Radians Chart

Following is a chart showing degree measures along with their corresponding radians. To make calculations easier and faster, we can also use this chart to convert degrees to radians. We can see from the chart below that 0 degrees equals 0 radians, and 360 degrees equals 2π radians.

how convert degrees radians formula equation conversion
how convert degrees radians formula equation conversion

Define Degree

Degrees are used to measure plane angles, also called arc degrees or degrees of arc. The angle measure for a complete rotation is 360 degrees, denoted by the symbol (°). To measure an angle in degrees, a protractor is used, which measures angles in degrees by measuring angles in 360 degrees.

define degree
define degree

Formula for converting Degrees to Radians

A generalized formula for converting degrees to radians is used to convert angle degrees into radians. The angle in degrees can be converted into radians by multiplying it by π / 180. 

First, one degree is equal to π /180 radians & one radian is equal to 180/ π radians. 

You can convert a specific number of degrees in radians and multiply the number of degrees by π / 180.

Derivation

A complete counterclockwise revolution equals 360 degrees and as can be seen in the image below, a complete counterclockwise revolution equals 2 π in radians. In other words, these statements are equivalent to:

  • One full counterclockwise rotation in degrees = 360 degree
  • One full counterclockwise rotation in radians = 2 π

Degrees to Radian Equation

Circles with radian measures of 2* and degrees of 360 subtend angles at their centers. Here is an equation that represents the relationship between degree measure and radian measure:

  • 360 degree = 2 π radian
  • 180 degree = π radian

By using this equation = 1 degree = π/ 180 radians & the above formula can be used to make conversions between degrees and radians, and vice versa, if necessary.

Degrees to Radian Conversation

Angles are measured using degrees and radians as units of measurement. Any angle measured in degrees can be converted to radians according to our convenience. There are a few very basic calculations that can be done to make this conversion.  Let’s define both units and establish the relationship between them.

Write down the degree measure of the angle.

A degree is equal to (π)/180 radians. We multiply the angle given in degrees by π /180 degrees to convert it to radians.
Angle in Radians = Angle in Degrees x π /180 degree.

Express the answer in radians by simplifying the values.

Value of 1 Radian

A radian value can be expressed in terms of radian measure and a degree value can be expressed in terms of degree value. By simply assuming the value of π as 22/7, we can write the equation as follows: 

value of 1 radian
value of 1 radian

1 Radian = 180 degree / π

1 Radian = 180 degree / 3.14

1 Radian = 57 degree 19 minutes approx

Additionally, 1 degree = π / 180 = 0.01745

Convert 45 degrees in Radian

Angle in Radians = Angle in Degrees x π /180 degree.

= 45 * π /180

= π / 4

Angle in Radians = π / 4

Convert 100 degrees in Radian

Angle in Radians = Angle in Degrees x π /180 degree.

= 100 * π /180

Angle in Radians = 1.745 Radian

Convert 12 degrees in Radian

Angle in Radians = Angle in Degrees x π /180 degree.

= 12 * π /180

= π / 15

Angle in Radians = π / 15.

Radian to Degree Conversation

In previous sections, we discussed how to convert degrees to radians for different angles. Now let’s see how we can convert radians to degrees for any particular angle. Radians can be converted to degrees using the following formula:

Radian * 180 / π = Degrees

By using this formula we can convert π / 3 radians into degree.

Or, π / 3 * 180 / π = 60 Degrees.

Convert 5 π / 9 radians to degrees

Radian * 180 / π = Degrees

By using this formula we can convert 5π / 9 radians into degree.

Or, 5 π / 9 * 180 / π = 100 degrees

Answer: 5 π / 9 radian = 100 degrees.

Convert 7 π / 36 into degrees

Radian * 180 / π = Degrees

By using this formula we can convert 7π / 36 radians into degrees.

Or, 7π / 36 * 180 / π = 35 degrees.

Answer: 7π / 36 radian = 35 degrees.

Negative Degree to Radian

In the same way that we have converted positive degrees into radians, we can do the same with negative degrees. By multiplying the given angle in degrees by π /180, you get the angle in degrees.

If -180 degrees must be converted into radians, then:

Angle in Radians = Angle in Degrees x π /180 degree.

= – 180 * π /180

Angle in Radians = – π

Convert -72 degree to Radian

Angle in Radians = Angle in Degrees x π /180 degree.

= – 72 * π /180

= – 2π /5 radian

Answer: – 72 degree = – 2π /5 Radian.

Convert – 99 degree to Radian

Angle in Radians = Angle in Degrees x π /180 degree.

= – 99 * π /180

= – 11 π / 20 radian

Answer: – 99 degree = – 11 π / 20 radian.

Specialty of Pie

Pi is the most famous number in the world. Pi, or the ratio of the circumference to the diameter of a circle, appears to be a straightforward concept. The exact value of the number is inherently unknown because it is an irrational number.

It is not possible to only measure radians in Pi, they are just numbers. The radius of a circle is equal to the ratio of its length to its radius. Radians are 2PiR divided by R if the arc goes around 360 degrees or a full circle. 360 degrees are equal to 2 Pi radians.

Reason behind Using Radian instead of Degree

Radians allow linear measures and angle measurements to be related. A unit circle has a radius of one unit. A radius of one unit is the same as a circumference of one unit.

In radians, we can measure angles and write down formulas for arc lengths and areas of sectors in circles.

In addition to linear measurements, radians are also used to relate angles and linear measurements. The radius of a unit circle is one unit. 

One unit along the circumference corresponds to one unit along the radius. Start with zero at (1, 0) and wrap a counterclockwise number line around it.

Why Do You Consider Angle Measurements in Radians?

So far, the angle unit of measure has been degrees. Radians are another unit you should be familiar with. 

It is important to note that the circumference of a circle is equal to 2πr, where r is the radius. A circle’s radius and circumference can be divided apart to give us simply 2π: this is the distance around its center independent of its radius. 

Alternatively, you can say 2π is the angle swept by the radius of the circle. It should be noted that 2π is also equivalent to 360 degrees. In this case, the radian is the unit of angle. The radius of a circle is equal to the angle formed such that the portion of the circle swept by the angle equals the radius. The angle formed by the radii connecting the center of the circle to the endpoints of the arc would have a measure of 1 radian if we took a segment of length r and molded it onto the circle. A radian is approximately equal to 57.3 degrees.

In accordance with convention, the angle θ is measured counterclockwise from x-axis. A negative angle value is assigned if the clockwise direction is used. The same point on the circle corresponds to both positive radians (180°) and negative radians (–180°).  

Initially, we focus on angles between – 2π and 2π, inclusive (– 2π ≤ θ ≤ 2π).
You can define 2π in angles larger in magnitude, but it can overlap with a lower value angle, so we neglect it.

Mathematical Problem of converting degrees to radians

A nearby store sold Katie a pizza. Several slices of pizza were given to her by her mother, one measuring (1/6)th of the complete pizza. Her mother asked Katie about the angle formed by the central slice of pizza. How do you measure the angle in degrees and radians for Katie?

Katie’s portion represents the (1/6th) of the entire pizza. Pizza has a circular shape, and a complete circle has an angle of 360 degree, so we can conclude that:

Complete Pizza Angle = 360 degree

The angle for the (1/6) th size = 360 * 1/6 = 60 degree

Angle in Radians = Angle in Degrees x π /180 degree.

= 60 * π /180

= π / 3 radian.

Answer: Its angle is 60 degrees. It takes π / 3 radians to convert Katie’s slice from degrees to radians.

A and B on a circle with center O are two points on the circle where Angle AOB = 105 degrees. Now can you Convert angle AOB’s angle measure from degrees to radians?

You have angle AOB = 105 degrees

Angle in Radians = Angle in Degrees x π /180 degree.

= 105 * π /180

= 7 π / 12 radians

Answer:  Angle AOB =7 π / 12 radians.

Radian or Degree: which one is bigger

There is a huge difference between a degree and a radian. There are 2 π radians in a circle, which is slightly more than six radians. Almost one sixth of a circle is a radian, which is 57 degrees.

Merits of Radian

Radians are the most natural way to divide a circle because they are the natural measure for it. 

In order to go around a circle completely, you would need just over six arcs that lie on the circumference. Every circle has this property.

Radians are almost always used to express rotational motion equations. A problem’s initial parameters may be in degrees, but they should be converted to radians before being used. When describing a physical picture or measuring angles with a protractor, you should use degrees.

Demerits of Radian

It is a pure number since the radian measure is a ratio of two lengths. In mathematics, most ratios do not have a unit attached to them, although some add the unit radians to indicate angular measurements or length along a circle.

Radians are equivalent to meters of arc length per meters of radius. Due to the fact that the meters cancel out, it is dimensionless. A similar canceling without a circle or factor of 2 results in resistivity is being measured in Ω∙m.

FAQ

Which one easy: Radian or Degree?

Degrees and radians are both units for measuring angles. It’s pretty easy to convert radians to degrees. To begin, remember that 180 degrees is equal to π radians, or half a circle’s degree. Thus, one radian equals 180 degrees divided by π.

Can you determine radians in a triangle?

Euclidean space converts the sum of triangle angles into the straight angle, as in 180 degrees or radians, or as if the triangle was half-turned. An angle at each vertex of a triangle is bound by a pair of adjacent sides.

Why are degrees not Radians?

Despite the fact that radians are dimensionless, they can still be confused with Hertz although you do not want to confuse them. In contrast to 360 degree rotations, radians provide a much more natural description of angles.  

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