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 .
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.
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.
The area of a circle is give by one-half multiplied by τ (tau) mutliplied by the radius of the circle squared.
The circumference of a circle is given the constant τ (tau) multpilied by the radius of the circle, where τ = 2π.
The volume of a sphere is given by two-thirds multiplied by the circle constant τ (tau) multiplied by the radius cubed.
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, 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.
The circle constant appears in many advanced formulas such as the normal distribution, fourier transform, and more. These are shown below:
Euler's Formula returns the point on the the unit circle in the complex plane when given an angle.
The normal distribution is a probability density function that forms a bell curve shape.
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