Pi Day

Happy Pi Day! Today’s date, expressed at M/DD, comes out to an approximation of the value of pi—3.14. And if you celebrate the day at 1:59:26 (and forgive the ‘p.m.’), then you’re doubly a geek. More interesting information and links can be found in the Wikipedia article, or in the LA Times Web Scout blog here.

Today’s date expressed in DDMMYYYY format (14032008) occurs 9,209,525 digits after the decimal point in the value of pi.

I am going to make pie for dessert tonight. It will be ten inches (25.4 cm) in diameter at its base, and therefore, my slide rule informs me, 31.4 inches (79.8 cm) in circumference, since circumference of a circle is its diameter times pi. It will be 78.5 square inches (about 506 square cm) in surface area at its base, since surface area of a circle is calculated by multiplying the square of its radius (half its diameter) by pi. But, you ask, how much pie is there really? This is a bit more complicated and involves trigonometry (gasp!).

To find the volume of our pie, we have to go back to our high school math and remember our geometry and trig. This calculation is a bit trickier since the pie’s edges are (mostly) slanted, making it a frustum of a cone. To calculate the volume, then, we simply subtract the volume of the smaller from the larger cone, leaving the volume of the pie as the difference. The formula for the volume of a cone is one third of its height (from the tip to the centre of the circle at the base), times the square of its base’s radius, times pi. However, we don’t know the height of the cones, since these cones are only imaginary (in the metaphysical, not the mathematical, sense). Visualize the cones, in cross-section, as a triangle. The radius of the base of the smaller triangle is 5.00 inches, and we can use a protractor to find out that the base angle is 70 degrees exactly. Trigonometry tells us that the ratio of the two legs of a right triangle is given by the tangent of the angle opposite, so taking the tangent of 70° and multiplying by 5.00 gives us the height, which is 13.74 inches. This means that the volume of the smaller cone is 359.71 cubic inches. Because angles cut by a transversal on the same side of the transversal and parallel lines are congruent, the angle of the base of the larger triangle (the top of the pie) is also 70°. My ruler shows that the depth of the pie tin is 2.5 inches, so assuming the pie rises and is baked properly, we can use 2.5 inches as the height of the frustum. Add it to 13.74 from the height of the smaller cone and we get a total height of 16.24 inches. We now find the tangent of 70° and divide the total height by it, since we want to find the length of the radius of the base of the triangle. This yields a radius of 5.9 inches across the top of the pie, which allows us to calculate a volume of 592 cubic inches for the larger cone. Subtract the volume of the smaller cone and we get a volume of 232 cubic inches, or about 3800 cubic centimetres.

I will post a picture of this insane amount of pie in all its tasty glory once the baking is complete. (Blueberry filling, I don’t think I mentioned yet.) Whee!