Simulation · Maths · Class 11

What one radian looks like

From the lesson Radian measure and arc length in Trigonometric Functions. Change the values and watch what happens.

The idea behind it

NCERT §3.2.2

In a circle of radius 1 unit (a unit circle), the angle at the centre made by an arc 1 unit long measures 1 radian. The same holds in any circle: an arc as long as the radius makes 1 radian at the centre.
The whole circumference of a unit circle is 2π, so one complete revolution is 2π radians, a half turn is π radians and a quarter turn is π/2 radians.
An arc twice as long makes twice the angle at the centre, so the angle is proportional to the arc. An arc of length l in a circle of radius r therefore makes θ = l/r radians at the centre; equivalently l = rθ.
l = rθ holds only with θ in radians. For example, with π = 22/7, an arc of a circle of radius 21 cm making 60° = π/3 at the centre has length 21 × π/3 = 7π = 22 cm.
A radian is a ratio of two lengths, so it has no dimensions; the word radian is often left out, and an angle written without a degree sign is read in radians.
Radians match the real numbers: lay a number line along the tangent at a point A of the unit circle with 0 at A, and wrap the positive half anticlockwise and the negative half clockwise round the circle. Each real number x then lands at the point reached by an angle of x radians.

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