PiUsing a compass, draw a circle. Take one piece of string and place it on top of the circle, exactly once around. Now straighten out the string; its length is called the circumference of the circle. Measure the circumference with a ruler. Next, measure the diameter of the circle, which is the length from any point on the circle straight through its center to another point on the opposite side. (The diameter is twice the radius, the length from any point on the circle to its center.) If you divide the circumference of the circle by the diameter, you will get approximately 3.14. A larger circle will have a larger circumference and a larger radius, but the ratio will always be the same.

Relationship with Pi

The circumference of a circle relates to one of the most important mathematical constants in all of mathematics. This constant, pi, is represented by the Greek letter π. The numerical value of π is 3.14159 26535 89793 and is defined by two proportionality constants. The first constant is the ratio of a circle's circumference to its diameter and equals π. While the second constant is the ratio of the diameter and two times the radius and is used as to convert the diameter to radius in the same ratio as the first, π. The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. While the constant ratio of circumference to radius also has many uses in mathematics, engineering, and science, it is not formally named. These uses include but are not limited to radians, computer programming, and physical constants.

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ. As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ° is used. A complete revolution is 2π radians (shown here with a circle of radius one and thus circumference 2π). It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.