Introduction: Draw Pi(e) With Triangles, Like Archimedes
Here's a fun guide to finding your own pi. We're going to use a technique first put forth by the greek mathematician (scientist, physicist, inventor, etc), Archimedes of Syracuse. This guy used infinitesimals to figure out values, and we're going to use them to make a circle from smaller and smaller equilateral triangles... and then estimate pi with some simple trig. Fun!
Archimedes died for his circles. Roman soldiers invaded Syracuse, found Archimedes, who said, "Noli turbare circulos meos." Meaning "Do not disturb my circles." He was stabbed there and then.
Step 1: Gather Supplies
You'll need a fine tipped pen or pencil, a ruler and a piece of graph paper (any paper will do, but the graph helps you get a nice right angle when you start).
Step 2: Draw a Diamond
We'll start with the biggest triangles that make sense.
Draw two equal-length perpendicular lines intersecting in the middle. The intersection point will be the centre of your circle and each line coming from the centre shows the circle's radius.
I made my circle have radius 8cm. So, I drew 16cm lines, meeting at 8cm (aka the middle).
Now connect the end points to make four right-angle isosceles triangles. These form your perimeter, or "circumference", although it's not really a circumference at all right now. In fact, it doesn't look anything like a circle right now... but we'll get to that, so hang on.
Step 3: Cut Each Triangle in Half and Complete
On the four perimeter lines you just drew, mark their middle point. Now, draw new lines of length radius from the centre and passing through the middle point of the perimeter lines. Connect the end points of the new lines to the nearest radius points. This creates eight new isosceles triangles. (ie. you should make eight new perimeter lines for this step.)
Sweet, we're getting closer to pi!!!
Step 4: Keep Halving and Completing
Continue the process of halving each newly drawn perimeter line and drawing a line of length radius from the centre and passing through the point, then connecting all the radii you've drawn to create new isosceles triangles.
Step 5: Make It a Pie
I ended up with 32 triangles. It starts getting a bit messy and hard to see after that, and I think it looks fairly circular at this point.
After I finished this, I decided to give it a pie crust by making right-angle triangles from the radii. So, I drew a line from the 0/360˚ point to the 90˚ point (actually, you've already done this already... never mind). Then I drew a line from the 11.25˚ point to the 101.25˚ point, then 22.5˚ to 112.5˚ and so on until I had all 32 right-angle triangles drawn.
Step 6: So What's Pi?
Pi is circumference divided by diameter. We know the diameter (16cm for me), so what's the circumference? It's the sum of all the bases of the isosceles triangles.
The base of one isosceles triangle can be calculated using trigonometry (admittedly, something Archimedes didn't have, but this makes it easier... if you want to do it like Archimedes, use a ruler).
So, we know sin(angle/2) = (base/2)/radius and we know that the angle is going to be 2*360˚/n, where n is the number of triangles in the circle. So the base is: base = 2*radius*sin(360˚/n).
Now the circumference is n*base, which we can expand as n*2*r*sin(360˚/n).
So let's find pi with n=32.
pi = circumference/diameter
pi = [ n*2*r*sin(360˚/n) ] / 16
pi = [ 32*2*8*sin(360˚/32) ] / 16
pi = 3.14 ... Ta-da!!
While this doesn't really show Archimedes calculating pi (since sine functions needs pi) it does show his use of infinitesimal portions to calculate values. If you do the same calculation on n=4, you'll get something like pi = 2.83. Way off, but the thinner (more infinitesimal) your triangles get, the closer to pi you get... and the more your shape looks like an actual circle!!! Fun!!!
Step 7: Credit
Jason Padgett's How To Make a Fractal inspired me to make this, check out his art, it is beautiful.
Also, Vi Hart has a video about another way to draw circles with fractals and infinitesimals (as well as the proof of pi) that helped me plan this Instructable out.