Derivation of Π

The mathematical constant Π (pi) can be derived in many different ways, but here I will present my favorite. The method was first described by none other than the mathematical genius, Archimedes, who lived in the 2nd century B.C.!!
The knowledge contained herein requires none other than an understanding of basic high-school mathematics. Archimedes’ method is so genius because it is so simple. I will take you step-by-step in this derivation, using geometric figures as a reference. Here we go!

First, what is Π?

Π is a mathematical constant that seems to show up in a variety of seemingly unrelated areas of mathematics, although it is best described as the ratio between the circumference and the diameter of a circle. In other words:

Π = Circumference/diameter


How can we calculate the value of Π using Archimedes’ method?

Archimedes envisioned the fact that as the number of sides (n) of a polygon increases, the more it approaches the configuration of a circle. Imagine increasing the sides of a polygon from 4 (a square), to 5 (a pentagon), to 6 (a hexagon), to 8 (an octagon), and so on, to infinity. Therefore, a circle can be thought of as a polygon with an infinite number of sides. Even a polygon with 32,000 sides will appear to the naked eye to look like a circle, although it is not. Thus, if we can calculate the perimeter of a polygon with a large number of sides, we can therefore approximate the circumference of the corresponding circle. Knowing the circumference, calculating the value of Π is a piece of cake! Let’s try this method using a hexagon (a polygon with 6 sides), and then we can derive a general formula to approximate Π using polygons with any number (n) of sides.
Here is a hexagon:

Now, let us circumscribe a circle around this hexagon:

I suppose you can now envision that as we increase the number of sides (n) of this polygon, the perimeter of the polygon approaches that of the circumference of the circle.



Now, let us draw 2 lines, both starting from the exact center of both the circle and hexagon. These two lines will both end at two consecutive vertices of the hexagon. The two lines are equal in distance. We will call both of these lines “r”, as they also represent the radius of the circle:

Now, let us draw a line from the exact center of both the circle and hexagon to a point that bisects this side of the polygon. We will not name this line, as it will not be used in any calculations, but it will help you envision what will come next.
Notice that this new line creates 2 new right triangles which are equal in all corresponding sides, angles, and in area.
In bisecting this 1 side of the polygon, the new unnamed line also creates the right sides of the 2 triangles (i.e. the right side of the triangle is perpendicular to the bisecting line). With this bisection, we have also divided 1 side of the hexagon into two equidistant halves. We will call each of these halves “a”:

Thus, 1 side of the hexagon = 2a.



We subsequently describe the angle immediately opposite “a” as angle “x”:


Is there a way to determine the value of angle x? We can calculate the value of angle x as follows:

The larger angle 2x is simply equal to 1/6th of the complete 360° revolution of the entire hexagon, or 60°. Therefore, angle x is equal to 30°.


Now, let us assume that the radius (r) of this circle is equal to 1/2 (you will see why later). We can then describe the relationship between angle x, a, and r (1/2) using trigonometry (remember SOH-CAH-TOA!!) as follows:

This shows that r is equal to 2a, and therefore, the triangle created by the two sides r and 2a (2a is equal to 1 side of the hexagon) is an equilateral triangle, with all angles of this triangle also equal to 2x (60°).


Using the solved equation above for “a” and “2a”, we can determine the perimeter of the entire hexagon:

Remember that the perimeter of the hexagon is 3 when we assumed that the radius of the circle “r” was equal to 1/2. Note: of course this works no matter what we assume r to be, making r = 1/2 just makes Π easier to calculate. For example, if we assumed r = 1, then the perimeter of the hexagon would have been calculated as 6 instead of 3. All other proportions could have been considered, and the end result would always be Π.

Recall also that Π is defined as the ratio between the circumference and the diameter of a circle. The diameter of a circle is simply twice the radius:

Diameter of circle = 2r

Therefore, in utilizing Archimedes’ derivation of Π, an approximation of this mathematical constant can be determined by taking the ratio of the perimeter of the hexagon (which through this method approximates the circumference of our circle) to the diameter of the circle (2r):

Not a bad estimate of Π, considering we are only using a polygon with 6 sides (n = 6). As the number of sides (n) of the polygon increases, we will get closer and closer to the “true” value of Π. But can we derive a general formula that will approximate Π for a given value of n? Read on.



Let us go back to our circle and hexagon:

Remember how we calculated the angle 2x = 60°? We said that it could be easily calculated as 1/6 of the entire rotation of the 6-sided hexagon (1/6 of 360°). Let’s generalize this calculation:

Let us carry this calculation out to solve for 2a:

Dividing each side by 2:

Remember the following relationship (using trigonometry):

Nevermind what sin x is equal to, and assuming r = 1/2 again:

Remember that 2a is equivalent to one side of a polygon with n number of sides. Another way to say this is that 2a multiplied by n will give us the perimeter of a polygon with n sides. Thus, multiplying each side of the equation by n:

That’s it!! The formula to calculate the perimeter of any polygon with n number of sides (assuming r = 1/2).

Let’s try to calculate Π for a polygon with n = 10,000 sides (I’m not going to attempt to draw this polygon, but it has MANY more sides than our hexagon! To the naked eye, it would probably look like a circle):

perimeter = 3.141592601912665692979346419289

For radius = 1/2 (i.e. diameter = 1):

We have calculated Π correct to 30 decimal places!! Of course, using calculus, the “true” value of Π can be calculated with this method, assuming a polygon with an infinite numbers of sides n, which is really a circle. This sets up a limit, but that would require 11th grade mathematics, and I need to brush up on that a bit! Maybe for another time…..

Residency is a Killer

Been a loooooonnnngggg time since I hit this blog, got much to say but ain't felt like saying it. It's the damn residency I tell you, but one day I hope to get motivation back and start cranking out more insights and theories. My most qualitative writings depend immensely on what mood I'm in.....lately I've been focusing my utmost concentrations on nuclear physics and atomic weaponry, and I'm also putting the loose ends together on my medical knowledge, internal medicine is starting to make sense to me now and I'm good enough now where I can reasonably know what to do when somebody's sick. But the hours are too much, I got too much to do besides medicine!! J-mart has been in the renaissance period of his life for the past 5 years....but medicine is detracting from all the other scientific and creative areas he could master.
Here's a typical 4-day rotation on a typical medicine service with call---after you see this, you'll understand why this blog hasn't been active lately:


Tuesday, 6 AM: My alarm clock starts blaring in my ears....damn, time for yet another 30-hour shift, does this EVER END?

Tuesday, 7:15 AM: Rollin' into work, already exhausted....look up all the patients, go to morning report at 8 AM, rounds start at 8:30 AM, 14 patients, I'm leading my interns as we go room to room. Luckily everything went well today without a hitch.

Tuesday, 12 Noon: Rounds end, just enough time to roll down for noon lecture at 12:15, grab some grub (actually, just a bottle of water, since my residency decided to go with nuttin' but healthy lunches--wheat bread sandwiches and vegetable wraps--disgusting!!).

Tuesday, 1:15 PM: Noon lecture ends, pretty dope topic on some core medicine, upstairs to wrap up the day's work.

Tuesday, 5 PM: Admissions start rolling in, between my 2 interns, I got 10 new patients to supervise, and around 40-50 old patients to cross-cover on all the 4 medicine teams, luckily my interns are tight and can handle the issues well.

Wednesday, 4 AM: FINALLY finish up the 10th admission, roll up to the call room to crash.

Wednesday, 6:30 AM: The alarm in my call room rings, it's time to get up and start checkin' the computer for labs and vitals and tests on all the patients on my team.

Wednesday, 8 AM: Morning report begins again, I'm beyond exhausted at this point, but still manage to participate some. I think the 2 1/2 hours of sleep I got actually made me feel more tired than if I'd just stayed up all night. I'm in some gross, sweaty scrubs and haven't showered in over 24 hours.

Wednesday, 8:30 AM: Rounds begin again. 16 patients this time around. Let's get through this as fast as we can.

Wednesday, 12 Noon: Rounds are finally over, I'm dragging at this point.

Wednesday 12:30 PM: Grand rounds begin, some lecture about a study on some G-protein receptors. Boring--get me home QUICK!

Wednesday, 1 PM: In the middle of Grand Rounds, I get up to leave, gotta be out of the hospital to comply with the 24 + 6 hour rule. 30 hours are over, but they seem to go by pretty quick, only about 40 more of these 30-hour shifts before I graduate residency!!

Wednesday, 1:15 PM: Open the door to my apartment, exhausted. I plop my a** down on the lazy boy and turn on the TV. Can't get up, even to eat.

Wednesday, 4 PM: Still haven't gotten up from that lazy boy, but I figure I can move over to the bed to go to sleep.

Wednesday, 11 PM: I wake up, restless, I try but I can't get back to sleep for a couple hours.

Thursday, 6 AM: My alarm awakes me again, since the beginning, I've done nothing but work and sleep since Tuesday morning. I'm excited I can finally take a shower, but I'm more exhausted this morning than I was yesterday. I call it the post-post-call depression.

Thursday, 7 AM: Repeat another day of work. I won't re-hash it, similar to what I did Tuesday and Wednesday.

Thursday, 5 PM: The work day is over, patients are tucked in, time to go home.

Thursday, 5:30 PM: Peeps is going out to eat and a movie, I take another shower and change my clothes.

Thursday, 6:15 PM: Roll up to the Westfire grille. Haven't eaten since my Tuesday call night. Really hungry, I order a couple drinks, some appetizer, chow it down quick, movie is starting soon.

Thursday, 7 PM: Walk over to the Cinemagic movie theater. We watch "The Covenant". The movie was OK, nothing special, but at least the main chick was really hot.

Thursday, 9 PM: Get home from the movie, tidy up some things, watch a little TV.

Thursday, 10 PM: Time for bed, tomorrow I will feel a little bit better than today, hopefully.

Friday, 6 AM: Alarm rings again, as usual, I throw my pillow across the room and start screaming--"This never ends!!"

Friday, 7:15 AM: Roll into work again, repeat the day per Tuesday, Wednesday, or Thursday, take your pick.

Friday, 4:30 PM: Finally get out of work again, today I had clinic in the afternoon, and everything went smoothly.

Friday, 6 PM: Just foolin' around, ready to go out tonight, but tomorrow I gotta get up early again for another 30-hour shift, so I gotta keep it early unfortunately.

Friday, 9:30 PM: Stumblin' in, luckily didn't drink too much, so tomorrow hopefully won't be that rough.

Friday, 10 PM: I get to bed, damn, it's time to repeat this entire 4-day cycle all over again!

Saturday, 6 AM: The alarm rings again! Here we go, another 30-hour shift, repeat the same 4-day cycle like I started on Tuesday morning, who knows what tonight will bring. Only gotta do this 5 more times this month. Next month will not be as bad as this one, hopefully.

Adding this up, between Tuesday morning and Saturday morning, the only free time I've had when I'm not working or sleeping was Thursday and Friday evening......