Radian and Degrees are the two interconvertible units used to measure angles having a specific relation, where radian is considered a dimensionless SI unit while the degree is a non-SI unit. The relation between radian and degree measurements is based on the fact that they both represent different ways to measure angles. Angles can be expressed in terms of radians or degrees, and there is a simple conversion factor between the two. Before getting to the detailed relation between the radian and degree measure, let us revise the radian and degree measure.

Radian Measure (rad)

The International System of Units (SI) describes the radian as an angular measuring unit. One radian can be stated as the angle formed at the circle’s center when the radius and length of the arc of that circle are equal. In other words, if you took a circle whose radius is ‘r’ and wrapped its arc around its center, the angle generated at the center would be 1 radian.

Further Explore: Explanation of Radian

1 radian pic

Degree Measure (°)

Degree is another unit of angular measurement commonly used in everyday contexts. A circle is divided into 360 degrees, with each degree representing 1/360th of a full circle.

Mathematical Relation Between Radian and Degree

Radians and Degrees are the units used for the angle measurement. They are interconvertible with each other. The relation between radians and degrees is given below.

\textbf{As we know that}

In One Complete rotation, there are 360 degrees

\text{Number of degree in one revolution}=360^o

Whereas,

\text{Number of radians in one revolution } = \frac{\text{Circumference of circle}}{\text{radius of circle}}
\text{Number of radians in one revolution }= \frac{ 2\pi r}{r}
 \text{Number of radians in one revolution }= 2\pi \text{ radians}
\textbf{We also know that}
    \text{Number of degrees in one revolution}=\text{Number of radians on one revolution}

Therefore,

360^o=2\pi \text{ radians} 
\text{  1 radian} = \frac{360}{2\pi}
\text{ 1 radian} = 57.3^o

An angle of approximately 57° corresponds to 1 radian.

Similarly, To determine the one-degree measure, use the above equation. i-e

360^o=2\pi \text{ radians}
\text{  1 degree} = \frac{2\pi}{360}
\text{ 1 degree} = 0.0174\ rad

Further Explore: Common And Important Questions Regarding Radian

Relation between radian and degree measure

Conversion Between Radian And Degree Measures

The conversion factor between radians and degrees is as follows:

\text{  1 radian} = \frac{180}{\pi}\text{degree}
\text{  1 degree} = \frac{\pi}{180}\text{radian}

Here, π (pi) represents the proportion between a circle’s diameter and circumference, which is approximately equal to 3.14159. So, if you want to express an angle in degrees instead of radians, it can be achieved by multiplying the radians value by (180 divided by π). Similarly, the conversion of degrees into radians is accomplished by multiplying the value of degrees by (π divided by 180).

Explore More: How many radians account for the circumference of the circle

conversion of radians into degrees picture

Formulas For The Conversion Between Radian And Degree Measure

A circle is divided into 360 degrees, where each degree represents 1/360th of a full revolution. On the other hand, a circle is also divided into 2π radians, where each radian represents 1/2π (approximately 0.3183) of a full revolution.

Degrees and Radians can be interconverted to each other using the following formulas.

For the conversion of degrees into radians: radians = degrees × (π/180)

For the conversion of radians into degrees: degrees = radians × (180/π)

We may build a relation between radian and degree using the above formulas.

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Formulas for the conversion of degrees into radians

Examples Of Radian and Degree Relation

For example, let’s assume you have a 45-degree angle. You would use the formula: radians = 45 (/180) 0.7854 radians to convert it to radians.

Similarly, Using the formula: degrees = 1.5 (180/) 85.94 degrees, you can convert an angle of 1.5 radians to degrees.

With the use of these formulas, it is possible to interconvert radians and degrees into each other and are helpful when working with trigonometric functions or when comparing angle measurements in different systems.

Some Common And Important Questions Regarding Radian and Degree Relation

During the study of a radian and degree relation, the below-mentioned questions are the most common and important questions that are asked in the examination and have major importance in the multiple choice questions. Therefore go through these questions attentively.

Que#1 Convert 0° to radian

There is a very simple conversion between the radian and degree. To convert 0° into radians we have two possible methods. Both of these methods are explained below.

1st Method

According to the mathematical relation between radian and degree;

\text{ 1 radian} = 57.3^o

By dividing 57.3 on both sides we get;

1^o = \frac{1}{57.3}

By dividing 1 by 57.3 we get the answer 0.017

   1^o = 0.0174\text{ rad}

To find 0 in radian, Multiply both sides by 0

   1^o(0) = 0.0174(0)\text{ rad}
   0^o = 0\text{ rad}

From the above calculation, it is clear that 0 degrees in radians is equal to 0 radians

2nd Method

There is also an alternate method to determine the radian and degree relation. This method is quite a simple conversion method. In this method conversion between the radian and degree takes place by using their conversion formula. To know about the formula check out the section ” Formulas for the conversion between radian and degree measure“.

By using formula

\text{radians}=\text{degrees}\times\frac{\pi}{180}

put 0 degrees in the above equation

\text{radians}=\text{0}\times\frac{\pi}{180}

\text{radians}=\text{0}

Hence for 0 degrees, we get 0 radians.

Que#2 What is 45 degrees in Radians

For the conversion of 45° into radians, there are also two methods, that are as under.

1st Method

For the derivation of the first method check out the section “Mathematical radian and degree relation

By using the relation, we know;

   1^o = 0.0174\text{ rad}

For the conversion of 45 into radians, multiply both sides by 45.

   1^o(45) = 0.017(45)\text{ rad}
   45^o = 0.783\text{ rad}

2nd Method

In this method, we use a formula for the radian and degree conversion. For the derivation of the formula, go through the section “Conversion between radian and degree.’

The formula for their conversion is;

\text{radians}=\text{degrees}\times\frac{\pi}{180}

Put degree = 45

\text{radians}=\text{45}\times\frac{\pi}{180}
\text{radians}=\frac{\pi}{4}

Hence 45 degrees in radian is π/4

Que#3 What is the radian equivalent of 90 degrees?

The radian equivalent of 90 can be determined in two ways. The first method gives the numerical value while the second method gives a fractional value of the radian and degree relation.

1st Method

Using Mathematical relation,

   1^o = 0.0174\text{ rad}

Multiply both sides by 90 to find the radian equivalent to 90 degree

   1^o(90) = 0.0174(90)\text{ rad}
   90^o = 1.56\text{ rad}

2nd Method

Using the conversion formula between radian and degree relation

\text{radians}=\text{degrees}\times\frac{\pi}{180}

put the value of degrees that is 90.

\text{radians}=\text{90}\times\frac{\pi}{180}
\text{radians}=\frac{\pi}{2}

Hence the answer to the question ‘What is the radian equivalent of 90 degrees’ is π/2

Que# 4 How do you convert 180 degrees to radian?

The following two methods can be used to convert 180 into radians.

1st Method:

This method of radian and degree relation is used to determine the numerical value of their conversion.

According to the mathematical relation

   1^o = 0.0174\text{ rad}

Multiply both sides by 180

   1^o(180) = 0.017(180)\text{ rad}
   180^o = 3.14\text{ rad}

2nd Method

This method of radian and degree conversion is used to determine the value of radians in terms of pi.

Using the conversion formula.

\text{radians}=\text{degrees}\times\frac{\pi}{180}

Put degree=180

\text{radians}=\text{180}\times\frac{\pi}{180}
\text{radians}={\pi}

180 degrees in radians is equal to π radians.

The conversion table of some common and important radian and degree relation is mentioned below.

The conversion diagram of the radian and degree relation is given below.

Frequently Asked Questions- FAQs

Q#1 How radians and degrees are related to each other?

Radians and degrees are related to each other as 1 radian corresponds to 57.3 degrees.

\text{ 1 radian} = 57.3^o

Q#2 What is 1 radian equal to in degree?

An angle of approximately 57 degrees corresponds to 1 radian.

 Q#3 What is 60 degrees in radian?

\text{60 degrees in radian is equaln to 1.047 or }\frac{\pi}{3}
\textbf{1st Method:}
   \text{ 1 radian} = 57.3^o
1^o = \frac{1}{57.3}
   1^o = 0.017\text{ rad}
60^o = 0.017(60)\text{ rad}
60^o = 1.047\text{ rad}
\textbf{2nd method: }

By Using Formula

\text{radians}=\text{degrees}\times\frac{\pi}{180}
\text{radians}=60\times\frac{\pi}{180}
\text{radians}=1.047\ rad

Q#4 How to determine the relation between radian and degree?

The radian and degree relationship can be determined with the help of the below formulas.

  • radians = degrees × (π/180)
  • degrees = radians × (180/π)

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