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?
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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.
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 $L_{crustsquare} > L_{crustcircle}$