Thread: another portion of dependence area of square to area of circle based on the same diag

1. another portion of dependence area of square to area of circle based on the same diag

dependence diameter of square to diameter of circle based on the same diagonal is ( 4 * d / sqrt(2) / d ) to ( pi * d / d )
area of square to area of circle based on the same diagonal is ( ( d / sqrt(2) ) ^ 2 / d ) to (( pi * d ) ^ 2 )

2. Re: another portion of dependence area of square to area of circle based on the same

That is so "terse" it is difficult to tell what you are saying.

A circle with diameter "d" has radius d/2 and so area $\displaystyle \pi r^2= \pi \frac{d^2}{4}= \frac{\pi}{4}d^2$.

A square with diagonal d (so the it is inscribed in the circle) has sides of length $\displaystyle \frac{d}{\sqrt{2}}$ and so area $\displaystyle \frac{d^2}{2}$.

The ratio of "area of circle to area of square" is $\displaystyle \frac{\frac{\pi}{4}d^2}{\frac{d^2}{2}}= \frac{\pi}{2}$.
And the ratio of "area of square to area of circle" is $\displaystyle \frac{\frac{d^2}{2}}{\frac{\pi}{4}d^2}= \frac{2}{\pi}$, the reciprocal, of course.

But that doesn't appear to have anything to do with what you wrote. I can't make anything out of "4*d / sqrt(2) / d". What do the two "/" mean? If that was supposed to be $\displaystyle \frac{\frac{4d}{\sqrt{2}}}{d}$, that is equal to $\displaystyle \frac{4}{\sqrt{2}}$. The "d"s cancel and that certainly is not the area of the square. And why is the $\displaystyle \pi$ squared in "(pi * d)^2". That is NOT the area of a circle of diameter d.

3. Re: another portion of dependence area of square to area of circle based on the same

Hello, razera!

Sloppy wording . . . I had to guess what you meant.

We have this scenario:
Code:
              * * *
*           *
o - - - - - - - o
*|             * |*
|           *   |
* |         *     | *
* |       o       | *
* |     *  d      | *
|   *           |
*| *             |*
o - - - - - - - o
*           *
* * *

dependence diameter of square to diameter of circle based on the same diagonal is:
. . . ( 4 * d / sqrt(2) / d ) to ( pi * d / d ) . Why did you divide by d?

You want the ratio of the perimeter of the square to the circumference of the circle.

The side of the square is: $\displaystyle \tfrac{d}{\sqrt{2}}.$ .Its perimeter is: $\displaystyle 4\left(\tfrac{d}{\sqrt{2}}\right) \,=\,2\sqrt{2}d$

The circumference of the circle is: $\displaystyle \pi d$

The ratio is .$\displaystyle 2\sqrt{2}d: \pi d \;=\;2\sqrt{2}:\pi$

area of square to area of circle based on the same diagonal is:
. . . ( ( d / sqrt(2) ) ^ 2 / d ) to (( pi * d ) ^ 2 ) . no

The side of the square is: $\displaystyle \tfrac{d}{\sqrt{2}}.$ .Its area is: $\displaystyle \left(\tfrac{d}{\sqrt{2}}\right)^2 \:=\:\tfrac{1}{2}d^2$

The radius of the circle is: $\displaystyle \tfrac{d}{2}.$ .Its area is: $\displaystyle \pi\left(\tfrac{d}{2}\right)^2 \:=\:\tfrac{\pi}{4}d^2$

The ratio is .$\displaystyle \tfrac{1}{2}d^2 : \tfrac{\pi}{4}d^2 \:=\:2:\pi$