Slicing Pi

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hat good is a whole pi anyway?
Pi is the ratio between the diameter of a circle and its circumference ...... π = 3.14159...
π/4 = .7854... , π/6 = .5236... are however much more useful in many problems.

For instance, what is formula for the area of an ellipse?
Answer: .7854 x (the 'normal box' that contains it)

Note: Normal means perpendicular to the major and minor axis.

enclosed ellipse

That means any ellipse whatsoever. A circle is .7854 times the square that contains it.

πr2 = .7854... (when r = 1/2)
Meaning that the side of the box containing it is 1 unit.
Now, if we stretch the box (uniform deformation), all the little increments (dx) within the square are all deformed in like fashion and therefore the relationship of the ellipse to a square is the same as the relationship of a circle to a square. Same, same all same.

Wouldn't this number be easier to memorize than some hairy formula (in which you would need the length and width of the ellipse anyway)?

It gets better in 3d.

A sphere is .5236... x (the normal box that contains it). That's π/6

4/3 π r3 = volume of sphere

If r = 1/2 as before then, 4/3 π 1/2x1/2x1/2 = π/6

Now we can uniformly deform the box as before into numerous shapes,

ellipsoids
prolate spheroids
oblate spheroids
And ... even egg shapes!

Like this, cut the sphere and the box in half and stretch the two halves nonuniformly so that one side is dumpier than the other.

stretch the box and make a new form

Then, stretch the top half and make an even stranger form. What is the volume of the resulting shape?

Complicated formula? ... no ... still the same ...

.5236 x (the size of the box that contains it)

Why?

Because the two halves are perfectly symmetric, it doesn't matter how we stretch them ... the relationship in each side is still the same ... .5236.

So just put them back together and measure the sides of that box.



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