Maximise pizza crust

Question

You are baking a pizza but you love the crust and so want to maximise it. Is it better to bake a circle or a rectangular pizza - assume both must be the same thickness?

There is no limitation on the shape of your baking tray.

My guess is the rectangular one is better. But can someone help me prove it?

If you're allowed to make a star, go for it! Or a rectangle of thickness at most twice the width of the crust for an all-crust pizza. — Arthur, Aug 14 '17 at 12:20
The circle is the worst possible shape for a crust-lover. Any other shape you can bake will be better. — David K, Aug 14 '17 at 12:28
@Arthur von Koch snowflake shaped pizza for the win! Infinite crust! — Xander Henderson, Aug 14 '17 at 15:21

score 4 · Accepted Answer · answered Aug 14 '17 at 12:23

4

The obvious answer is to make a rectangular with small width and large length. This way you will only get crust.

But now say you want to chose between a square pizza and a circle pizza. I assume the volume of the two pizza to be the same (you eat the same amount of food) and the thickness of the pizza to be the same.

For the circle : $$V_{pizza}=\pi r^2h$$ $$L_{crustcircle} = 2\pi r = 2\pi \sqrt\frac{V}{\pi h}= 2 \sqrt\frac{\pi V}{h}$$ For the square : $$V_{pizza}=a^2 h$$ $$L_{crustsquare} = 4a = 4 \sqrt\frac{V}{h}$$ Since $2 \sqrt\pi < 4$ you get $L_{crustsquare} > L_{crustcircle}$

answered Aug 14 '17 at 12:23

stity

3,500

That's cool, thanks. I answered it for square in a bit different way. – user471851 Aug 15 '17 at 14:50
1

I would have called $a$ the thickness and $z$ the radius. It is known that $V=\pi zza$ – zwim Oct 31 '20 at 03:33

score 0 · Answer 2 · answered Aug 15 '17 at 15:26

Reducing the problem to compare crust length of square (sq) vs circle (cl) pizza.

Since both pizzas use the same amount of dough and have to be the same thickness, the surface area of both pizzas will be the same. (In fact, the surface area of any pizza shape - even irregular - rolled out from the same amount of dough to the same thickness will be the same.)

So, $S_(sq)=S_(cl)$ ⇒ $$a^2 = \pi r^2$$ $$\frac{a}{r}=\sqrt\pi = 1.77$$

Now, if the perimeter crust of the square pizza (P) is longer than the circumference of the circle pizza (C), $\frac{P}{C} > 1$.

$P = 4a$ and $C = 2\pi r$

$$\frac{P}{C} = \frac{4a}{2\pi r} = $$ $$= \frac{2}{\pi} * \frac{a}{r} = 0.637 * 1.77=1.13$$

So, $\frac{P}{C} >1 $, which means you get more crust from a square pizza.

Maximise pizza crust

2 Answers2