Halving a slice of pie

By Abhijit Menon-Sen <ams@toroid.org>

2009-08-26

A few days ago, I bought a slice of Date and Apple Pie from Eatopia.

(Eatopia is a food court at the India Habitat Centre in New Delhi. It is noisy and crowded, but used to have a pretty good bakery. I haven't been there for some years, but Hassath and I happened to be in its vicinity, so we stopped in to pick up a sandwich, a croissant, and a slice of the pie that was once a particular favourite of my father's and mine.)

Hassath was going to eat the sandwich; the croissant was mine. The sandwich was larger, so I finished my croissant first, and was reaching for the pie when a thought struck me: where exactly should I bite it to get no more or less than my fair share?

The slice was too wide to fit in my mouth sideways, so I couldn't try to bite it in half lengthwise (precious crumbs!). I would have to approach this resource-sharing problem pointy-end first, and bite very carefully.

Thinking quickly, I simplified the pie slice to a circular section (assuming that it had uniform thickness, and giving up on the crusty outer edge). It was an eighth of the pie, so its area was πr²/8, and the angle at the vertex was π/4 radians. My fair share (ignoring, in the interests of simplicity, the fact that I clearly deserve a larger piece for forgoing the crust) would thus be an isosceles triangle with half that area; and its height is what I needed to determine.

The area of an isosceles triangle with height l and base d is l×d/2. We know that is equal to half of πr²/8; and we can also express d as 2l×tan(&pi/8), π/8 being half of the central angle. Thus 2l²×tan(π/8) equals πr²/8, and so l is the square root of πr²/(16×tan(π/8)); in other words, l is r times some constant, which suits us fine.

tan(π/8) gave me a bit of pause, before I remembered that π/4 was a more tractable angle, and tan(θ) equals sin(2θ)/1+cos(2θ). sin(π/4) and cos(π/4) are both equal to 1/√2, so the required tangent is √2−1 ≅ 0.4142. Losing patience, I simplified progressively: 0.4 times 16 is 6.4, which is about twice π, so l ≅ r/√2 ≅ 0.7r. I stuffed the pie into my mouth and bit off a piece that looked about right.

So much for applied math. The pie was awful.

(After I'd finished eating, I realised—looking at the remaining piece—that I had incorrectly assumed that my bite mark would be a straight line. If, instead, I had incorrectly assumed that it would be a section of a circle concentric to the outer edge, I could have saved myself some trigonometry and the answer would have been exactly r/√2. But the pie wouldn't have tasted any better for it.)