Trigonometry: radians

[ radians ][ arc length ][ sector area][ small angles ]





What is a 'radian'?

radian definition

A radian is the angle subtended at the centre of a circle by an arc the same length as the radius of the circle.


1C (meaning 1 radian)= 57.296 deg.


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Arc length

radian definition #2

The arc length is proportional to its subtended angle.

Hence, if θ(theta) is in degrees and 'l' is the arc length:

equation for arc length


An angle can be expressed in radians by dividing the arc length by the radius.

Therefore θ in radians is given by:

arc length

Therefore for a circle(a 360 deg. angle), where the arc length is '2πr' and the radius is 'r' , the number of radians is 2πr/r , i.e. 2π .


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Sector area

The area of a sector is proportional to the angle its arc subtends at the centre.

If a sector contains an angle of θo then its area is given by:

sector area

However, if θ is in radians, remembering there are 2π radians in a circle:

sector area #2



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Small angles

small angles

For small angles(<10 deg.) there is a convergence between the value of the angle in radians with the value of its sine & tangent.

This approximate sine value may be expressed as:

sine theta approximates t o theta

The approximate cosine value is obtained thus:

small angle - cosine equation





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