One radian is the angle θ swept along a circle whose arc length S is equal to the radius R of the circle.(1)

Remember that π is the ratio of the circumference C of a circle to its diameter, which is twice its radius R, i.e. π = C / 2R.

From this we know the circumference C of a circle is 2πR, i.e. C = 2πR.

We also know C/R = 2π (2)

To find the number of radians in a circle, we divide the circumference C by the number of times we can fit the arc of length R from (1) around the whole circle, but that is just C/R, and that is always (2). So, there are 2π radians in a circle.

For any angle θ in radians, θ is in terms of the radius R and arc length S: θ = S/R

When arc length of S = length of R, then θ = 1 radian.

There are always 2π radians in a circle.

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