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
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 --
-- 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.
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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.
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.
Remarkably enough, as we will see, the theorem applies both to the boundaries and the areas:
If C1, C2 are the circumferences of any two circles, and D1, D2 their diameters, then
been called π, then
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,
For when we prove that the area of a circle is
-- then we have one of the most remarkable theorems in all of geometry:
The circumference of a circle is to the perimeter
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:
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
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.
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Copyright © 2016 Lawrence Spector
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