Pi day! (3/14/09)

[fluffy]

I'm only using circles and squares though, both equilateral shapes. To put it another way:

(c=circumference, r=radius, a=area, p=perimeter)

Yes, your argument is crap. Just because two shapes are both equilateral doesn't mean they are at all equivalent in any way.

Think of a circle as a polygon with an infinite number of sides. Consider a 3x4 right triangle. Its perimeter is 3+4+5=12. Its area is 6. Compare this with a 2x3 square. Its perimeter is 4+6=10. Its area is 6.

How can the area of a circle as compared to the area of a square, using exactly the same circumference/perimeter not always be consistent? I'm not saying they should be the same as each other, but the relationship should always be consistent, but I've just demonstrated that it is not.

Different objects' circumference:area ratios scale differently. Also it's a polynomial relationship, not linear. That can mess up your intuitive sense of things.



As far as the exact numbers you're getting, I have a feeling you're making an egregious math error since they shouldn't be anywhere even remotely similar. Are you remembering to take the appropriate square roots etc.?

[fluffy]

circle: c 100, r 15.915494309189533576888376337251, a 628.3185307179586476925286766559

A circle with circumference 100 will have diameter 100/pi = 31.83, or radius=15.92, and an area of 795.77. How are you getting 628?

A square with circumference 100 will have side 25 and an area of 625.

[irwin]

How can the area of a circle as compared to the area of a square, using exactly the same circumference/perimeter not always be consistent? I'm not saying they should be the same as each other, but the relationship should always be consistent, but I've just demonstrated that it is not.

Ok, so you've got a circle of circumference c, having diameter 2r. It will have area a=pi*r^2

Now you've got a square, also of circumference c=4x. It has area x^2.

Assume: the area of the circle and the square are the same, as are their circumferences.

Therefore:

2*pi*r = 4*x
and
pi*r^2 = x^2

From the first equation, x=1/2*pi*r
By substitution,
1/4*pi^2*r^2 = pi*r^2

After cancellation (we assume r != 0), we are left with:
pi = 4

This is a contradiction, therefore the equations are inconsistent and have no solution, except for the trivial case of the area (and circumference) of both the circle and the square being 0.

[Märk]

circle: c 100, r 15.915494309189533576888376337251, a 628.3185307179586476925286766559

A circle with circumference 100 will have diameter 100/pi = 31.83, or radius=15.92, and an area of 795.77. How are you getting 628?

A square with circumference 100 will have side 25 and an area of 625.

Well there you go. I was going pi (r X 2) instead of pi (r^2). ("My math might be wrong, but..")
Still, learned some interesting things.

[Spud]

circle: c 100, r 15.915494309189533576888376337251, a 628.3185307179586476925286766559

A circle with circumference 100 will have diameter 100/pi = 31.83, or radius=15.92, and an area of 795.77. How are you getting 628?

A square with circumference 100 will have side 25 and an area of 625.

An equilateral triangle with circumference 100 will have side 33.3333. and an area of 481.1252
So much for the equilateral figures argument.

[Billy's Little Trip]

Here, let me figure this out. Move!











































Mincemeat?