Interactive Polar Coordinate System

This interactive demonstrates the Polar Coordinate System. In the polar coordinates, the position of a point is defined by its distance from the origin (radius) and its angle relative to the origin. The angle is measured in radians where one full rotation is equal to Tau (τ) radians, where Tau is a geometric constant approximately equal to 6.28.


The symbol Tau (τ) is magical and is worth explaining more. The unit of radians, by itself, measures the angle of rotation using the radius of a circle. This is demonstrated in the polar plot below, where lines are drawn at 1 radian, 2 radians, 3 radians, and so on. Where a radian is one radius’ length along the circumference of a circle.

You’ll notice that there is some left over after 6 radians. So how many radiuses (radii) fit around the circumference of a circle or rephrasing the question, what is the circumference of a circle divided by its radius? It turns out this number is approximately 6.2831853071.

Fortunately, mathematicians use the symbol τ (tau) to represent this number. This simplifies the polar coordinate system and makes it easier to represent angles. For example, one full rotation is and half a rotation is .

You may be wondering about the number π (pi). π is an equivalent way to represent an angle in radians, except π represents a half-rotation instead of a full rotation. See The Tau Manifesto.