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Degree And Radian Measure


What is a Degree Measure?

One degree is one 360th part of a full circle. To get a more clear idea we define degree measure as follows:

A central angle that is subtended by an arc equal in length to 1/360 of the circle's circumference, is said to have a measure of one degree, denoted  1°.

That is; for a circle with circumference C units, central angle of θ degrees subtended by an arc of s units, this relationship can be expressed by the following proportion:


maths_a2_15The degree is further divided in to 60 minutes.

For even finer measurements the minute is divided again into 60 seconds; however this last measure is so small, it only used where angles are subtended over extreme distances such as astronomical measurements, and measuring latitude and longitude.

What is a Radian Measure

maths_a2_05The radian is the standard unit of angular measure. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.

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. The radian is a fixed size no matter what the size of the circle is.

Important Remarks:

  • A full angle is 2Π radians, so there are 360° per 2Π radians, equal to 180° / Π.
  • A right angle is Π / 2 radians and a straight angle is Π radians.

maths_a2_16Relation between Degree and Radian

A circle subtends at the centre an angle whose radian measure is 2Π and its degree measure is 360°,

Radian to Degree

We have,

2Π radian = 360° or Π radian = 180°

Also, 1 radian = 180° / Π

Or, Radian Measure = maths_a2_08  x Degree Measure

Degree to Radian

We have,

1° = Π / 180 radian
Degree Measure = maths_a2_09x Radian Measure


On the basis of above discussion, we now solve the following questions:

1. Convert 40° 20’ into radian measure.

We know, 180° = Π radian

40° 20’ = 40 maths_a2_12   degree = maths_a2_08  x  maths_a2_13 radian = maths_a2_14  radian.

2. Convert 6 radians into degree measure.

We know, Π radian = 180°

6 radians = maths_a2_09  x 6 degree = maths_a2_07  degree

343maths_a2_06   degree = 343° +   maths_a2_01 minute [As 1° = 60’]

343° + 38’ +   maths_a2_11 minute [As 1’ = 60’’]

343° + 38’ + 10.9’’’

343°38’11’’ approximately

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