Wut?
Everyone encounters Pi in their life. You know, 3.14 something something. No one really tries to figure out Pi, it’s just one of those numbers that exist. I’ve been lurking the internet, so I can see if I can incorporate it into my work, and stumbled onto this site (http://physicsinsights.org/pi_from_pythagoras-1.html)… Which made me realize why I did so poorly in school.
I have no idea what’s going on… Even after solving the problem!
It’s mostly because there are too many fancy letters, which makes the process of explaining unnecessarily complicated to people that aren’t a calculator.
His whole idea, is everyone’s idea of solving Pi: You have a shape (polygon) and if all the sides are equal (regular), you can probably make a circle inside of it that. As you increase the number of sides, the shape becomes more circle like. As Sides approaches infinity, the shape becomes a circle.
Now for the math.
Since we’re making a circle in a shape, that means the shape is bigger than the circle. The shape’s area will be greater than the circle’s area that we’re drawing. Therefore, if we’re solving Pi via Area, then Pi will always be overestimated.
Unless of course, we have an infinite number of sides, then the circle becomes the entire shape we’ve drawn!
The formulas are this:
The circle’s Area is
Pi*radius (or maybe radius^2? not sure, not bothering to check)
Lets make it simple and say the circle’s radius as 1, so it solves my dilemma above.
then
Area(circle) = Pi
Now lets talk about the shape. We need to know the shapes area based on the number of sides. There are many ways to do it, but it’s a matter of google and subsitution, and you get:
½ n Csc[(90 ° (-2 + n))/n]^2 Sin[(360 °)/n]
Area of a Square, with the inner circle’s radius as 1, is 4.
Area of a regular polygon with 100 sides, and the inner circle’s radius as 1, is 50 Sec[(9 °)/5]^2 Sin[(18 °)/5]… Or in english 3.14
Lets review,
As the shape increases the number of sides, the shape becomes more circle-like. Therefore, the shape’s-area, becomes more like a circle’s-area. Therefore, with the Area = Pi formula above, and keeping the circle’s radius = 1, you get a pretty good approx of Pi.