Trigonometry

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9

THE CIRCLE

A CIRCLE is a plane figure bounded by one line, called the circumference, such that all straight lines drawn from the center to the circumference are equal to one another.

A straight line from the center to the circumference is a called a radius.  A diameter is a straight line through the center and terminating in both directions on the circumference.

A radius, then, is half of a diameter; equivalently, a diameter is twice a radius:

D = 2r.

The definition of π

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

Any two quantities of the same kind have a ratio to one another, a relationship with respect to relative size. The circumference C and

diameter D of a circle are both lengths. And so they have a ratio to one another. That ratio -- how many times longer C is than D -- is called π.

C = πD.

Since ancient times, it was known that π is a bit more than 3. In the

  third century B. C., Archimedes of Syracuse proved that π is less than 3 1
7
,
  but more than 3 10
71
.
3 10
71
 < π < 3 1
7
.

As modern decimals:

  3.140845 < π < 3.142857

It was not until the 18th century that it was proved that π is irrational.  And when the ratio of two quantities is irrational, then we have no choice but to approximate it as a rational number.

C = πD3.14159 D.

It should be intuitively clear that π cannot be rational, because it signifies the ratio of a curved line to a straight. And to name such a ratio exactly is impossible.

In the next Topic, we will see how to approximate π.

In any case, the definition of π --

C = πD

-- becomes 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 π3.14.

To see the answer, pass your mouse over the colored area.  To cover the answer again, click "Refresh" ("Reload").

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.

How do we know that the circumference of every circle has the same ratio π to its diameter?  The following theorem assures us.

Circles are to one another as their circumscribed squares.

(Euclid, XII, 2.)

Remarkably enough, as we will see, the theorem applies both to the boundaries and the areas:

1)   Circumferences of circles are to one another as the perimeters
of their circumscribed squares
.
 
  (The boundaries are proportional.)
 
And:
 
2)   Areas of circles are to one another as the areas
of their circumscribed squares
.
 
  (The areas are proportional.)

Now, the side of a circumscribed square is equal to the diameter D.  Therefore its perimeter is 4D.  Algebraically, then, statement 1) is:

If C1, C2 are the circumferences of any two circles, and D1, D2 their diameters,

C1 : C2 = 4D1 : 4D2 .
 Therefore alternately,
 
C1 : 4D1 = C2 : 4D2 .
This means that all circumferences have the same ratio    C 
4D
 to the
  perimeters of their circumscribed squares.  And since the ratio   C
D
 has

been called π,

 C 
4D
  =   π
4
 .  .  .  .  .  .  .  .  .  .  .  . (1)

A circumscribed square

π
4
 thus signifies the ratio of the circumference of any circle to the

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.)

Next, statement 2):

If A1, A2 are the areas of any two circles, and D1, D2 their diameters,

and therefore alternately,

This means that all circles have the same ratio   A
  to their

circumscribed squares.

We are about to see that  π
4
 may be more fundamental than π itself.

For when we prove that the area of a circle is

A =  π
4
D²,
  so that   A
  is also equal to  π
4
 --
A
  =   π
4
,    compare line (1),

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

 C 
4D
  =   A

A circumscribed square

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.

That ratio is called  π
4
.

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 area of the circle will be a bit more than three fourths of the area of the square.

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

π
4
  =  1 −  1
3
  +   1
5
  −   1
7
  +   1
9
  −    1 
11
  +   .  .  .

If we took just the first two terms of that series, that would tell us that the circle is approximately two thirds of the square.  If we took three terms, we would know that the circle is appoximately thirteen fifteenths of the square.  Each term brings us a little more and then a little less than the actual ratio that the circle has to the circumscribed square.  But we

  will never be able to name that ratio exactly.   π
4
 is an irrational number

If there is anything in mathematics that deserves to be called beautiful, it is here.  We find such beauty especially when geometry is reflected by simple arithmetic.  The Pythagorean theorem is another example:  3² + 4² = 5².  We discover those relationships in those archetypal forms. We do not invent them.


Next Topic:  The area of a circle


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