This presentation illustrates the relation between radian and degrees

1. Relation between radian and degrees
by
SBR
www.harekrishnahub.com

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• Consider a circle with centre O and radius
r units.
• Let PQ be the diameter of the circle.
• Let A and B be any points on the circle,
such that the length of the arc AB is equal
to the radius r of the circle.
• Then the angle ∠𝐴𝑂𝐵 will be equal to 1
radian. (i.e., ∠𝐴𝑂𝐵 = 1 𝑐
)

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We have, length of the semi-circular arc = 𝝅𝒓 and the length of the arc 𝑨𝑩 =
𝒓
∠𝑨𝑶𝑩 = 𝟏 𝒄
and ∠𝑷𝑶𝑸 = 𝟏𝟖𝟎° = 𝒙 𝒄
(say)
We know that in a circle, the arc lengths are proportional to the angles
subtended by them at the centre. Therefore,
𝒂𝒓𝒄 𝑨𝑩
∠𝑨𝑶𝑩
=
𝒂𝒓𝒄 𝑷𝑸
∠𝑷𝑶𝑸
𝒓
𝟏 𝒄 =
𝝅𝒓
𝒙 𝒄
𝒙 𝒄
𝟏 𝒄
=
𝝅𝒓
𝒓
= 𝝅
∴ 𝒙 = 𝝅 𝒄
but 𝒙 = 𝟏𝟖𝟎°
∴ 𝟏𝟖𝟎° = 𝝅 𝒄

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In practise, the subscript for radian is usually omitted.
It is therefore, understood that
∴ 𝝅 𝒄 = 𝟏𝟖𝟎°
𝝅 =
𝟐𝟐
𝟕
= 𝟑. 𝟏𝟒𝟐
∴ 𝟑. 𝟏𝟒𝟐 𝒄 = 𝟏𝟖𝟎°
𝝅 ⟹ 𝟏𝟖𝟎°

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some examples:
Degrees Radians
360 𝟐𝛑
270
𝟑𝛑
𝟐
180 𝛑
90
𝛑
𝟐
60
𝛑
𝟑
45
𝛑
𝟒
30
𝛑
𝟔