For a circle, the circumference divided by diameter equals pi, which is the transcendental number 3.1415… Is there anything interesting about a similar type of ratio for polygons? If you divide the circumference of a polygon by it’s “diameter” (the distance from corner to opposite corner), does it produce some sort of remarkable or interesting number like it does for circles?
As the number of sides of your polygon approaches infinity, the side length of the polygon approaches pi (times twice your radius). You can define the radius of a polygon by finding the midpoint (where the distance to each vertex is equal) and then measuring that distance.
This we call the “apothegm”; and also, if you take the “radius” as being to the vertex, the ratio between twice this and the perimeter tends to pi from the other direction as the number of sides increases.
There is a ratio, and not a very friendly one, for any given number of sides and the “radius” or the apothegm, but pi is the only one that’s widely applicable enough that anyone cares about it.
For every n, there is some ratio between the perimeter of an n-sided polygon and its “diameter”.
If we consider the diameter to be twice the distance from a corner to the center (which will be the distance between opposite corners when n is even [so that “opposite corners” exist]), this will be sin(a half turn/n) * n.
If we consider the diameter to be twice the distance from a side to the center (which will be the distance between opposite sides when n is even [so that “opposite sides” exist]), this will be tan(a half turn/n) * n.
π will always be inbetween these two. As n grows larger, the former ratio increases and the latter ratio decreases, with π being the asymptotically limiting value of both. [In fact, this is essentially how Archimedes calculated π by inscribing and circumscribing polygons around a circle, calculating these bounds for successively doubled values of n using the (equivalent of the) trigonometric half-angle formulae]
I plan to to think about OP really really hard tonight seven seconds before 9:27.
I hope you thought about a triangle a few minutes before, then a square, then a pentagon, faster and faster…then, afterwards, thought about the same things in reverse order, at a gradually slower pace.
Nitpick - this is only true for ***Regular ***polygons and classic geometry. For real shapes - the answer depends on the scale being used.
For example - the perimeter of Hawaii or the perimeter of a spilled droplet of water on the floor totally depends on the scale you are using and can approach infinity. IIRC - you can have a finite area enclosed by an infinite perimeter - fractal theory.
What…the…f. 
Math humor, having to to with the number values for the “pi” of other shapes. Just guffaw and nod.
I suppose this will spoil the coy fun, Chessic Sense, but Leo Bloom is referring to the fact that yesterday (3/14/15 9:26:53.58979… etc.) the date and time matched the decimal digits of π. JKellyMap is referring to the fact that the perimeter that the sequence of constants noted in post #4 start out below π (with the perimeter-to-“corner-based diameter” ratio, which is lowest for a 3-sided figure, and increases as the number of sides increase from there), and end up above π (with the perimeter-to-“side-based diameter” ratio, which is highest for a 3-sided figure, and decreases as the number of sides increases from there).
OK, here’s a relationship: If we define the “radius” of a regular polygon as the distance from the center to the center of one of the sides, and call the ratio of the perimeter to the radius 2*pi[sub]n[/sub], then we’ll also have that the area of the polygon is pi[sub]n[/sub]*r[sup]2[/sup], just like with the circular pi.
My wife and I were watching the movie Interstellar last night when the power went out (due to thunderstorm). I don’t know remember how it happened, but we got discussing the fact that PI equals the ratio of the Circumference to Diameter of a circle and how remarkable that this is true of every circle.
Then, one of us (don’t remember which), asked, “Say, for a regular polygon, is there a similar number that is the ratio of the Perimeter to… the “diameter”…?”
We were both relieved when we were able to figure out the answer with our high-school math. For a polygon with n sides (let n = any positive whole number > or = 3), there is a formula for the ratio of P / D, where P is the Perimeter and D is the “Diameter”, which we define to be twice the “Radius” and where we define the Radius to be the distance from the center of the polygon to any one of its vertices. Of course, for a regular polygon, the terms “Distance” and “Radius” don’t both have exactly the same meaning as in a circle. The Radius seems correct enough since, as we’ve defined it, it’s the same length as the Radius in a circle which circumscribes the polygon in question (i.e., goes through all the vertices). But the Diameter concept in a polygon is… “off the map”, shall we say, until n gets very large (approaching a circle).
At any rate, the formula is as follows: P / D = n * Sin (pi / n).
As n -> infinity, the ratio P / D approaches pi. It never becomes pi, so long as n is finite, but it gives as good an approximation as you have time to calculate.
Try it out. If you have access to a spreadsheet (like MS Office’s Excel or Open Office’s Calc), put 3 in the top left cell (called A1). Then, in column B, put this formula:
=(A1) * SIN(PI() / A1)
Now, fill the numbers in column A1 with positive whole numbers for as many rows as you like. Drag the formula in B1 down in parallel. Viola! After about 10 rows (the “decagon”), you get to an approximation of pi that will look familiar… 3.14159.
It’s difficult to understand the derivation without a picture but, alas, I am new and don’t see how to post a picture.
I didn’t see the reply from Indistinguishable (posted earlier). My post is only half as good as that post - and more than twice as long! But thanks, Indistinguishable, for the interesting information about Archimedes.
For my wife and me, it seemed natural to take the Diameter to be twice the center to one of the vertices rather than twice the center to the midpoint of one of the sides, because the Radius has the “natural” meaning that it has in the circle when we use it this way. But it is interesting that it occurred to Archimedes to take advantage of the ambiguity to prove something interesting.
ERRATUM: I was just pulling from memory. Actually, you get a good approximation if you put in n = 1000. At n = 10, you only get 3.09017. I got 3.14159 (five digits of accuracy) at row 822 just now, by running out all the rows from 3 through 1000. This is interesting, because the eye ceases to tell the difference between a circle and a regular polygon at about n = 20 (“dodecagon”?). But it isn’t until a regular polygon of 822 sides that we get a ratio of Perimeter to “Diameter” that is recognizably pi.
Glad you enjoyed it!
What is the “natural” meaning? Both are quite natural extrapolations from the circular case, just in opposite directions, so to speak: For a regular polygon, the corners are the furthest points from the center, and so we might think of them as giving the largest “radius”, while the midpoints of the sides are the closest points to the center, and so we might think of them as giving the smallest “radius”. Of course, with a circle, all points are equally far from the center, and so the largest and smallest radius coincide.
Pi is transcendent but the ratios for regular polygons wouldn’t be, would they?
@ Indistinguishable:
You are right.
To me, it seemed natural to circumscribe rather than to inscribe. However, for a given regular polygon, it is arbitrary as to whether we make a circle inside the polygon which touches all the midpoints of the sides or we make a circle outside the polygon which touches all the vertices of the polygon. And so, the radius of the inner is as good a choice as the radius of the outer. At any rate, I cannot see any argument for my choice going beyond preference.
Good point.
Is it possible to calculate the circumference of a polygon without using pi or the trig functions which are associated with pi (like sin)? I can see it’s natural to use the trig functions since the circumference of the polygon can determined by dividing it into triangles, but then it’s not surprising that the answer is similar to pi since pi is an integral part of the calculation. If you didn’t have pi and the trig functions, how would you determine the circumference of the polygon?
Like, the Pythagorean theorem (a^2 + b^2 = c^2) can be used to calculate the length of 3rd side of a right triangle provided you know the length of the other two. Pi is not involved in that equation. Are there equations like that which could be used to solve for the circumference of the polygon? The answer, of course, will still be similar to the trig method, but this way it wouldn’t have pi as part of the actual equation.
The inner and outer ratios for regular polygons will not be transcendental; as noted above, these will be tan(a half turn/n) * n and sin(a half turn/n) * n, respectively; these live in the ring generated by adding to the rationals a primitive n-th root of -1 (i.e., a primitive 2n-th root of unity), and accordingly will be roots of polynomials (with integer coefficients) of degree at most n [we can be more specific about the degree with more care, but nevermind that for now].