Overview
The number represented by the Greek letter pi (3.14159…) is one of the most famous irrational numbers in mathematics. It is the ratio of the circumference of a circle to its diameter. The number represented by the Greek letter tau (6.283185…) is the ratio of the circumference of a circle to its radius. It is another irrational number that has many uses in mathematics.
Many Uses of Pi
The constant pi is used in many different mathematical contexts. For example, the circumference of a circle C is equal to pi times the diameter d, and the area of a circle A is equal to piπ times the radius squared r2. Other common equations that are not as clearly related to circles also use the constant pi, such as the distribution of the normal curve, the period of a pendulum, and quantum mechanics.
Diameter and Radius
Many circular objects can be measured by their diameter. Suppose you have a coin, or a plate, or a CD. It’s easy to put a ruler across the circle to see if it will fit into a particular space. However, a circle is defined as the set of points from the center to its radius. Think of drawing a circle with a compass. The point of the compass is the center, and the radius can be made any size. The ratio of the circumference of the circle to its radius is tau. (In math symbol language C/r = τ.)
Angles and Degrees
The measure of the angle between any two points along the circle is the same no matter the size of the radius. That measure is usually represented by another Greek letter theta θ, and is represented by degrees. A circle has 360o in one turn around the entire circle, so that a 90o angle is a quarter of the way around the circle, an 180oangle is halfway around the circle, and a 360o angle is one full turn around the circle. All the angles are measures in between those parameters.
Radian Angle Measure
Think of a pair of concentric circles, circles with the same center but different measures for their radii, as each circle has a different measure for its radius. Another way to measure angles is by radians, or the measure of the proportion of the arc length s to the radius r. That proportion s/r is also the Greek letter θ. All angles also be measured in terms of their fractional part of a circle. This also means that the length of the arc can be redefined as the fraction f of the full circumference C. (In math symbol language θ= s/r = (fC)/r = f (C/r) = fτ.) Therefore angles can be defined as fractions of tau τ.
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