Is there a proof that the ratio of the circumference of a circle to its diamante is always the same.?

Question

I have looked at some of Euclid's proofs in his 13 books but can not find a proof.

Answer 1

The closest I could come up with is a precursor to Area = pi * r^2. Namely, Book XII, Proposition II, where it said that the area of a circle varies constantly with the square of the radius.

Answer 2

The previous answerer is correct that you can prove this by approximating a circle by polygons, and this type of argument was probably the way it was originally done. You can do it more quickly using calculus as it is understood today.

First, assume with out loss of generality that the circle is centered at (0,0), and say it has radius r. Then the circumference is given by:

c = 2*int_{-r}^{r} sqrt{1 + x^2/(r^2-x^2)} dx

Now if you change variables in this integral to x' = x/r, dx' = dx/r you get that the circumference is

c = 2*r*int_{-1}^{1} sqrt{1 + x'^2/(1-x'^2)} dx'

Thus c/(2r) = int_{-1}^{1} sqrt{1 + x'^2/(1-x'^2)} dx'

Since 2r is the diameter this shows that the ratio of circumference to diameter is always the same.

Answer 3

look up pi.
ratio of the circumference of a circle to its diamante is always the same. i.e = to pi.

Irrationality and transcendence
Main article: Proof that π is irrational
The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known. A somewhat earlier similar proof is by Mary Cartwright. See proof that π is irrational.
Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.
http://en.wikipedia.org/wiki/Pi

Answer 4

We didn't make up C = pi*D off the top of our heads, did we? You can prove it yourself from the limit of approximating a circle surface from equal-sided polygons.

Answer 5

Is it "DIAMANTE" or "DIAMETER"?