What is a Radian?
A radian is a measurement of an angle different from typical degrees, and directly relates an angle to the radius and circumference of a circle. One radian occurs where the arclength is equal to the radius of a circle. In a circle, the circumference is equal to \(2\pi r\), so the number of radians in a full circle is \(\frac{2\pi r}{r} = 2\pi\), around 6.28 radians.
\(\theta = \dfrac{s}{r}\)
Where:
- \(\theta\): the angle in radians
- \(s\): the arclength
- \(r\): the radius of the circle
Key values to know:
- \(1 \text{ rad} \approx 57.3°\) — arc length equals the radius
- \(\pi \approx 3.14 \text{ rad}\) — a semicircle (180°)
- \(2\pi \approx 6.28 \text{ rad}\) — a full circle (360°)
Interactive Radian Explorer
Drag the point along the circle to see how arc length builds angle
Current Angle
1.00
rad
(57.3°)
1 radian — arc length = radius
θ = s / r = 1.00
s = rθ = 1.00 r
Convert Between Radians and Degrees
Another helpful resource for understanding radians is when you need to convert from radians to degrees and vice versa.
- To convert from radians to degrees, you can use the formula: \(degrees = radians \times \frac{180}{\pi}\)
- To convert from degrees to radians, you can use the formula: \(radians = degrees \times \frac{\pi}{180}\)