Tau

τ (tau) circle constant

The circle constant (tau) is a number approximately equal to . The number is defined as the length of the circumference of a circle divided by the length of its radius:

The number appears naturally when measuring angles in radians. A full rotation in radians is equal to radians. For reference, a full rotation in degrees is . The signficance of the number tau comes from the fact that the distance required to travel the circumference of a circle is equal to tau multiplied by the radius of the circle.

What About (pi)?

π (pi) definition

The constant (pi) is defined as the length of a circle’s circumference divided by the length of its diameter. The constant is directly related to :

While is an adequate constant, has long-term advantages as the better constant. The circle constant makes talking about, teaching and converting the unit of radians easier. For these reasons, this website presents equations and concepts using the circle constant .

Formulas

Here are some geometry formulas that utilize the constant .

Area of Circle Formula

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

Area of Circle Formula
Circumference of Circle Formula

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

Circumference of Circle Formula
Volume of Cylinder Formula

The volume of a cylinder is equal to PI multiplied by its radius squared and its height.

Volume of Cylinder Formula
Volume of Cone Formula

The volume of a cone is given by one third multiplied by PI, the radius of its base squared, and its height.

Volume of Cone Formula
Volume of Sphere Formula

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

Volume of Sphere Formula

Links

Radians | Concept

Radians are a unit that measure angle 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.

Polar Coordinate System | Concept

The Polar Coordinate System describes points in space using an angle and radius relative to the origin.

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.