Radians are a unit that measure angles using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle. For example, the angle of 2 radians can be visualized by the three circles below. Geometrically, the angle’s arc-length along the circumference of the circle will be equal to the numbers of radians multiplied by the radius of the circle.
More generally, all possible angles from zero to a full-rotation can be visualized using the interactive of the polar coordinate system shown below. The position of each point (r, θ) is represented as a radius r (distance from origin) and an angle θ relative to the origin. Try dragging the blue control point around.
Observe that a full rotation in radians isn’t a perfect number. For each circle, the length of its circumference is equal to six radians plus a fraction of a radian. To be more scientific, a full rotation in radians is equal to the length of the circumference divided by the length of the radius of the circle. This number is independent of the size of the circle and its value is approximately .
Mathematicians use the symbol τ (tau) to represent a full rotation around the circle in radians. Similarily, the symbol π (pi) is used to represent a half-rotation around the circle. τ is equal to 2π.
Using the number τ (tau) in the polar coordinate system makes talking about radians easier. A full rotation around the circle is equal to 1τ (“one tau”), a half-rotation is equal to .5τ (“one-half tau”) and so on. This is demonstrated below.
There is another unit of measure for angles called degrees, where 360° is a full rotation. While degrees are a perfectly good way to measure angles, the number 360 is arguably less significant than the length of the radius of a circle relative to its circumference.
The result of having two units of measure is that calculators will often have a setting of one or the other. So it is useful and even required to know how to convert from one unit to the other unit.
Converting Degrees to Radians
To convert a number from degrees to radians multiply the number by divided by .
Converting Radians to Degrees
To convert an angle from radians to degrees multiply the angle by and then divide by radians.
Constants τ (tau) and π (pi)
τ (tau) is a geometric constant approximately equal to 6.283. The number is naturally occuring as any circle's circumference divided by its radius.
The greek letter π (pi) is a naturally occuring number that is defined by any circle's circumference divided by its diameter.