# Geometry: what is a radian

I was doing a geometry question and I had all the formula right but the one part I was stuck on was the concept radians please could someone explain it to me and difference between radians and degrees

The arc length an angle subtends in a circle is proportional to that angle. A radian is the angle for which the arc length is one radius. The circumference is $\pi$ diameters and $2\pi$ radii, so one complete revolution is $2\pi$ radians. Traditionally a circle was divided into 360 degrees since that number has a convenient list of factors, and so one radian is $180/\pi$ degrees.

• So the the formula is ?/360 or ?/180 – user437688 Apr 17 '17 at 19:14

Given an angle, using the vertex of the angle, draw a circle of radius "R". The radian measure of the angle is the length of the arc the angle cuts off the circle, divided by R.

Equivalently, using the vertex of the angle as center, draw a circle of radius 1. The radian measure of the angle is the length of the arc.

To find a relation between degrees and radians, consider a "straight angle". That is, a straight line thought of as an angle. In degrees, the angle has measure 180 degrees. A circle of radius r has circumference $2\pi$. Cut by that line, we have a semi-circle with length $\pi$. So 180 degrees corresponds to $\pi$ radians. Because the relationship between radians and degrees is linear, if an angle has degree measure $\theta$ then it has radian measure $\frac{\pi}{180}\theta$.

Simply radian is a unit for angle.

$180°=\pi$ radian.

So, $1°=\frac{\pi}{180}$ radian

Hence $x°=x\frac{\pi}{180}$ radian