The definition of π

The student no doubt knows a value for the famous number π — 3.14 — but that is not its definition.  What, in fact, is the meaning of the symbol "π"?

π symbolizes the ratio -- the relationship with respect to relative size
-- of the circumference of circle to its diameter, whatever that relationship might be.

So when we say that π = 3.14, we mean that the circumference of circle is a little more than three times longer than the diameter.

C D

=  π3.14

Meanwhile, since

then we use that as a formula for calculating the circumference of a circle:

Or, since D = 2r,

C = π· 2r = 2πr.

Problem 1.   Calculate the circumference of each circle.  Take π = 3.14.

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a)   The diameter is 5 cm.   3.14 × 5 = 15.7 cm

b)   The radius is 5 cm.   3.14 × 2 × 5 = 3.14 × 10 = 31.4 cm

Problem 2.    The average distance of the earth from the sun is approximately 93 million miles; assuming that the earth's path around the sun is a circle, approximately how many miles does the earth travel in a year?

C = π × 2r = 3.14 × 2 × 93 million = 584.04 million miles.

It is possible to prove that the area A of a circle is given by the formula,

To gain some insight into that, let us circumscribe a square about the circle.

Now the side of the square is equal to the diameter D.  Therefore we know that the area of the square is D².  The question is,

What ratio has the circle to the circumscribed square?

If we can answer that -- if we can express that ratio by some number k -- then the area of the circle will be

A = kD².

First, though, let us answer an easier question:

What ratio has the circumference of the circle
to the perimeter of the circumscribed square?

That is, what ratio has the boundary to the boundary?

Since the side of the square is equal to the diameter D, then the perimeter of the square is 4D.

And since, by definition,

then on dividing by 4,

perimeter of the circumscribed square.  And since π is a bit more than 3, we see that the circumference is a bit more than three fourths of that perimeter.

For if we could prove that the area of a circle is

-- then we would have one of the most remarkable theorems in all of mathematics:

The circumference of a circle is to the perimeter
of the circumscribed square, as the area
of the circle is to the area of the square.

Again, since π is approximately 3, then just as the circumference is a bit more than three fourths of the perimeter of the square, so the circle will be a bit more than three fourths of the square itself.

What is more, it is possible to prove the following:

to the square.

to the square, minus a third of the square.  And so on.  

If there is anything in mathematics that deserves to be called beautiful, it is here.  We find such beauty especially when geometry is reflected in arithmetic.  Moreover, we discover these relationships in those archetypal forms; we do not invent them.