Trigonometry

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11

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; or, equivalently, a diameter is twice a radius:

D = 2r.


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 the diameter.

C
D		   =  π3.14

π turns out to be a fascinating yet difficult number, because it indicates the ratio of a curved line to a straight.  In the next Topic, we will see how π can be approximated.


Meanwhile, since

C
D		   =  π,

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

C = πD

Or, since D = 2r,

C = π· 2r = 2πr.

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

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 have the same ratio 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 in the same ratio as the perimeters
of their circumscribed squares.
 
  (The boundaries are proportional.)
 
And:
 
2)   Areas of circles are in the same ratio 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
D²		   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
D²		   is also equal to  π
4		  --
A
D²		   =   π
4		 ,    compare line (1),

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

 C 
4D		   =   A
D²

A circumscribed square

The circumference of a circle is to the perimeter
of the circumscribed square, in the same ratio 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   π
4		   were equal to just 1, that would mean that the circle is equal

to the square.

If   π
4		   were equal to 1 −  1
3		 , that would mean that the circle is equal

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.


Next Topic:  Evaluating π


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