Τ (Tau)

Definition of the circle constant τ (tau)
Figure 1: τ (tau) Definition

The circle constant (tau) is a geometric constant that appears in numerous math formulas relating to circles and angles. The numeric value of is defined as the length of the circumference of a circle divided by the length of its radius and is approximately equal to [1].

Note: This website uses the constant (tau) instead of (pi) as the default circle constant[2]. The substitution can be used to translate between the two constants.

Usage

Radians

This figure illustrates the radian angle system annotated using the circle constant τ (tau).
Figure 2: Radian Angle System

In the radian angle system the circle constant is equal to a full rotation in radains. Measured angles are represented as fractions of the circle constant . For example, shown below are two angles measured using the circle constant.

This figure illustrates the standardized angle that represents an sixth turn.
Figure 3: 1/6 τ Radians
This figure illustrates the standardized angle that represents an fourth turn.
Figure 4: 1/4 τ Radians

Geometry Formulas

Here are some traditional geometric formulas in terms of the circle constant.

Area of Circle

Calculate the area of a circle given the radius.

Circumference of Circle

Calculate the circumference of a circle given the radius.

Volume of Sphere

Caclulate the volume of a sphere given the radius.

Volume of Cylinder

Caclulate the volume of a cylinder given the radius and height.

Advanced Formulas

The circle constant also appears in advanced applications such as the normal distribution, fourier transform, and more.

Links

References

  1. Approximate the Circle Constant
    Wumbo (internal)
  2. No, really, pi is wrong: The Tau Manifesto
    Michael Hartl