# Supplementary Topic

## The Value of p ...

This section of the supplementary topic is purely for your own joy and interest ... it is not needed to understand any of the relevant topics that are discussed.

The value of p = 3.14159265 ... which appears in defining the area of a circle and in measuring angles in units of radians is really no mystery. This value can best be thought of as the ratio of the circumference of a circle to its diatmeter. One can simply try to "draw" a perfect circle and measure the circumference and the diameter, and dividing the circumference by the diameter will give the value of p. It turns out that no matter how small or how big of a circle on chooses to draw, this ratio comes out to be a constant value that is roughly equal to 3.14159265 ... (within errors of measurement and drawing "perfect" figures and so on ...).

The above geometrical idea is the best way at this level to understand where the value of p comes from ... if you are inclined to think practically you can always imagine doing an experiment like the above and getting values for p and then trying to "perfect" your experiment.

Another way to think about the value of p is perhaps in terms of some of our trigonometric functions. If you look back at these functions you will realize that we defined the sine, cosine, and tangent functions with the geometry of a right triangle and we said that these were functions that applied to angles. In other words, we realized that once we have an angle we can take its sine, cosine or tangent and this gave us the ratio of the sides of a right triangle (which sides depended on which function we were looking at). Now consider a right triangle whose non-right angles are 45º (this is the nicest right triangle you can picture). This triangle will have both of its legs equal in length (since both of the non-right angles are 45º, and are equal).

The tangent of an angle is equal to the ratio of opposite to adjacent side in a right triangle. Thus, the tangent of 45º is equal to 1 since the opposite and adjacent sides in a 45º right triangle are equal in length.

If you recall from units of angles, a 45º angle is equal to p/4 radians. Thus:

tan (p/4) = 1

We can try to find the value of p by taking the inverse tangent (labeled Arctan) of 1 and multiplying it by 4. If you do this on your calculator you must be in radians mode (since the angle p/4 is being specified in radians):

p/4 = Arctan(1)
p = 4 x Arctan(1) = 3.14159265 ...

Actually ... there is a way to calculate Arctan(1) without a calculator and get the value of p, but you need not worry about it here.