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 **2π** (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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