Why does pi work for squares?

[ughaibu]

The distance from the centre of a chessboard, diagonally to a corner or orthogonally to a side, is four squares. So, a chessboard can be considered as a circle with a radius of four, with the radius measured by mobility of chess pieces not by general length. As a chessboard has four sides of eight squares, it's circumference is thirty two, so, the circumference (32) divided by twice the radius (8), gives a value for pi (4). This on it's own isn't interesting, however, this value of pi correctly calculates the area of a square, in the case of the chessboard, pi (4) multiplied by the radius squared (16), gives sixty four.

Is there a logical explanation for the consistent effectiveness of pi in both formulae when applied to squares considered as circles?

[ughaibu]

Thinking about this further, it occured to me that increasing the diameter by one would increase the circumference by the number of sides, so I tried using values of pi, equal to the number of sides, for triangles, pentagons and hexagons but, as the results require "hollow" areas, this doesn't appear to work.

The other feature, of a pi equal to four, in the case of a square, is that pi fractally divides the square into resultant squares whose area is the square of the original square's radius. So, these formulae begin to look sensible for squares, but in that case, why do they work with circles?

What I'm asking is whether this is just a coincident property of squares or if there is something interesting about this.

[ughaibu]

Another point that might be suggestive: for any value of pi, the ratio of area to circumference is such that if the area is equivalent to the radius, the circumference is two.

[Jay-qu]

Think about what your doing, by squaring the radius you are working out the area of one quater of the square and then multipling it by 4 to get the total area...

[ughaibu]

Jay-qu: Thanks, yes, I know (see post 2).

[Jay-qu]

:eek2: I see now, I wrote that after just reading the first post.

[ughaibu]

My fault, this was something I was thinking myself to sleep with last night, I should've had a decent think about it before posting.

My first comment in post 2 turns out to be nonsense, a pi equal to the number of sides works fine for triangles, pentagons and hexagons, if the initial polygon (radius of 1) is divided radially (this "radially" refers to dividing lines, not to "squares") from the centre of the area to the centre of the sides ie the sub-areas must have four sides. As this idea is concerned with dimensions of chess-piece-mobility, the irregularity of the "squares" doesn't matter, this is the point I'd overlooked, they only need be conceptually regular.