Differences Between Radian and Degree

An angle is a figure formed at the centre by two rays having common endpoints. Degrees and radians are the most commonly used measuring units to measure angles. A Degree is an angle made by one part of 360 equally divided parts of a circle at the centre with a radius of r. Radian is the angle made at the circle’s centre by an arc of length equivalent to its radius. 

Degrees and radians measure angles; however, they are two different concepts. This article attempts to understand the properties of degrees and radians and the relation between radians and degrees. Furthermore, we will also see the difference between a radian and a degree.

What is a degree?

A degree can be defined as the angle made by one part of 360 equally divided parts of a circle at the centre with a radius of r. The degree is calculated based on the rotation of the sun. Therefore, a degree is equal to the 1/360 angle of a full rotation of the sun. One degree is equivalent to 60 minutes, and a further one minute is equivalent to 60 seconds.

What is a radian?

A radian can be defined as the angle made at the circle’s centre by an arc of length equivalent to its radius. It is the ratio of the displacement length or distance travelled and the circle’s radius. It is an S.I. unit and can be said as the unit to measure the angles using the radius of the circle. The radian value of a complete circle is approximately 6.28 radians. 

What is the difference between degree and radian?

Degree and radian are different ways to measure the angles. Moreover, the properties of degrees and radians are pretty different. The difference between degree and radian are highlighted below:

1. A degree is an angle made by one part of 360 equally divided parts of a circle at the centre with radius r, whereas radian is the angle made at the circle’s centre by an arc of length equivalent to its radius.
2. A degree is an accepted S.I. unit, whereas radian is a derived S.I unit.
3. If a circle is divided into 360 equal parts, then 1 degree is equal to the 1/360 angle of a full rotation of the circle. Therefore, a complete circle is 360°, whereas radian is based on the circle’s radius. Hence, when an arc equal to the length of the radius is drawn along the circle’s circumference, and the angle made at the centre of the circle with the help of two straight lines with a common endpoint is connected to the end of the arc, the angle made is said to be equal to 1 radian. 
4. A degree can be converted into minutes and seconds, whereas radian cannot be further converted.
5. A degree is represented as °, whereas radian is represented as c.

What is the relation between radian and degree?

Both degree and radian are used to measure angles. They have wide applications; therefore, it is crucial to understand the relationship between degree and radian. Furthermore, we need to understand how degree and radian values are derived from understanding their relationship. 

 As,                                                     

Circumference of a circle = 2πr

(And a circle is divided into 360 equal parts                = 360° = 2πr 

 and one part is equal to 1°,so,360° is

 equivalent to its circumference)                  

                                                                                             = 180° = πr

( Now, radian is equal to the length of the radius = 180° = π × Radian

      here “r” in the equation.)                                      = 180° / π = Radian

                                                                                                 =180° / 3.14 = Radian 

                                                                                                 =57.33° = Radian.

Now that it is clear from the above equation how radians and degrees are interconnected, it is vital to understand how to convert degrees into radians and vice versa.

Degree in radian

A full revolution of a circle denotes 360°, therefore 

 = 360° = 2π × radian

 = 1° = 2π×radian / 360°

= 1° = π / 180°×radian

Now, the degree value can be converted into radian by multiplying the value by π / 180°.

 For example,

The value of 30° is equal to

 =30 × π / 180°

= π/6 radian

Radian to degree

 As,                                                     

Circumference of a circle = 2πr

(And a circle is divided into 360 equal parts                = 360° = 2πr 

 and one part is equal to 1°,so,360° is equivalent to its circumference)                  

                                                                                             = 180° = πr

(Now, radian is equal to the length of the radius = 180° = π × Radian here “r” in the equation.)                                                                    = 180° / π = Radian

The value of radian can be converted in degrees  by multiplying 180° / π.

For example, 

π/6 radian = π/6 × 180° / π

                 =180°/6

                 = 30°.

Conclusion 

Degree and radian are vital measuring units. Because of the properties of degrees and radians, they are used widely.  A degree can be defined as the angle made by one part of 360 equally divided parts of a circle at the centre with a radius of r, and a radian can be defined as the angle made at the centre of the circle by an arc of length equivalent to its radius. 

One degree is equivalent to 0.017453 radians, and 1 radian is equivalent to 57.2958°. Both degree and radian can be converted easily. To convert degree into radian, multiply the value of degree by π / 180° and to convert radian into the degree, multiply the value of degree by 180°/ π.