Trigonometry

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9

THE CIRCLE

The circle

The circle

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

The circle

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 more than 3 and, but less than 3 and.

3 and < π < 3 and.

In 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 for practial purposes we have to approximate it as a rational number.

C = πDapproximately3.14159 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 πapproximately3.14.

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.

The circle

A circumscribed square

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

The side of a circumscribed square is equal to the diameter D. Therefore its perimeter is 4D.  Hence the ratio of the circumference C to the perimeter is c-4d.  And since c-4d has been called π, then

 C 
4D
 =   π
4

The circle thus signifies the ratio of the circumference of a 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.)

We are about to see that The circle may be more fundamental than π. For when we prove the formula for the area of a circle, we will see that the area A has that same ratio to D2, the area of the circumscribed square.

 C 
4D
 =  A
D2
 =  π
4

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.

As the boundary is to the boundary, so the area is to the area.

We have:

A
D2
  =   π
4

Problem 3.   Express the area A of a circle as a function of

a)  the diameter D.    A =  π
4
D2
b)  the radius r.    A =  πr2,  upon replacing D with 2r.

Problem 4.   Calculate the area of a circle

a)  whose radius is 3 cm.    28.26 cm2

b)  whose diameter is 20 in.    314 in2

Finally, 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. The circle 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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