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].

Usage

The circle constant is used throughout mathematics. It appears in geometric formulas relating to circles and angles. The constant also appears in some suprising places too, such as the normal distribution. The circle constant makes talking about and using the radian angle system more straightforward.

Note: This website uses the constant (tau) instead of (pi) as the default circle constant for reasons discussed on this page. The substitution can be used to translate between the two constants.

Geometry Formulas

Here are some traditional geometric formulas in terms of the circle constant. While these may feel unfamiliar for those more comfortable with , these formulas expressed in terms of the circle constant are more consistent with the rest of mathematics and physics[2].

Area of Circle

The area of a circle is give by one-half multiplied by τ (tau) mutliplied by the radius of the circle squared.

Circumference of Circle

The circumference of a circle is given the constant τ (tau) multpilied by the radius of the circle, where τ = 2π.

Volume of Sphere

The volume of a sphere is given by two-thirds multiplied by the circle constant τ (tau) multiplied by the radius cubed.

Radians

When measuring angles in radians, a full rotation or “one turn” is equal to radians as shown in figure 2 below. This makes expressing radian angles as fractions of a whole more straight forward, simpler to convert degrees to radians and overall easier to talk about. Since radians are the preferred unit for measuring angles[3], using the circle constant instead of (pi) is an advantage for students first exposed to radians. In particular, since the trigonometric functions and the unit circle are expressed in radians for higher level math and physics.

Radian Angle System
Figure 2: Radian Angle System

Advanced Formulas

The circle constant appears in many advanced formulas such as the normal distribution, fourier transform, and more. These are shown below:

Euler's Formula

Euler's Formula returns the point on the the unit circle in the complex plane when given an angle.

Normal Distribution

The normal distribution is a probability density function that forms a bell curve shape.

references

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