Area of a circle and pi?

Question

Something that has had me curious since I did differentiation in my maths classes( a long time ago now!) is the equation for a circle ie x^2 + y^2 = 1 for circle with radius of 1. If you do the anti-differentiation for this equation, you can of course figure out the area of a circle. I'm curious whether by this anti-derivative you can show what pi is equal to. Is this in fact how pi was first calculated?

Answer 1

maybe.
the story goes that 'they' found out that the radius divided by the perimeter of the circle is a constant. independend of the size of the circle.

but ofcourse it could be as well that they found out that the area divided by the squared radius is constant.

the last one is less likely for me because its more complicated.

What I find interesting is that when a circle gets smaller and smaller it becomes a point, and that if you take certain complex line integrals around a singularity a point you will also see pi showing up in the answer.

see : http://en.wikipedia.org/wiki/Residue_(complex_analysis)

Answer 2

pi was first calculated using the inscribing, circumscribing technique.

assume a circle with diameter D.
you can construct a square circumscribing it by having four perpendicular tangents. the side will be length D, so the area of the square is D^2.

Now inscribe a square into the circle. This will consist of two right angle trangles sharing a common diameter. if the side is d, we get d^2 + d^2 = D^2, ie, d^2 = (D^2)/2

So we get the inequality (D^2)/2 < area of circle < D^2

Repeat this with a regular hexagon, and you'll get a tighter inequality. The larger the number of sides, the closer the limits of the area of the circle. So pi can be calculated by using, say an 18 sided regular polygon and doing this sort of calculation.

Answer 3

Bill X is right about the integral that you'd have to solve to get the area of the circle by this method. I think that this is one of those expressions that looks algebraic, but the integral turns out to be a trigonometric function (like arc cos x, for example). And I don't remember how to do that integral. Anyway, when you evaluate the trigonometric expression, it involves pi. But of course that doesn't give you a numerical VALUE for pi.

Anyway, pi was being worked on by the Greeks, long before calculus was invented, so the approach you describe is certainly not the FIRST way pi was approximated.

Answer 4

Hi. Pi was first calculated by taking a circle of 1 unit diameter and drawing polygons of ever increasing size inside and decreasing size outside. They approach pi. Done to infinity they would EQUAL pi (and each other).

Answer 5

To do the anti derivative of the circle equation you need to isolate for y, and that gives us y=+/-sqrot(1-x^2), and to take the anti derivative of that?! All I have for you is Good luck.