Should Pizzas be Square

Agreed. This a special first world problem that doesn’t require pressing global attention. Read something else. Goodbye.

You’re still here.

Okay then.

What is the most efficient pizza for the consumer? How would you define an efficient pizza? Assuming all other factors like taste, amount and type of topping, thickness of crust, presence or absence of cheese, oregano, freshly-shed snakeskin etc. equal, lets look at the following.

Area of Toppings (AoT)

Area of topping-less Base (AoB)

Again. Assuming all else equal, one might define an efficient pizza as one with maximum AoT and minimum AoB given the understanding

Area of Pizza, AoP = AoT + AoB

AoT is the good stuff. The part that hits the spot. The genie in the bottle. AoB is the sad limitation. The necessary evil. The Bottle. (Leaving aside its function as a grip or a handle, of course)

The parameter to be optimised for an efficient pizza is therefore, and unequivocally, AoT. Fair?

Great.

Now, AoB is a direct function of the perimeter or the circumference of the pizza surface. It’s easy to observe empirically that the larger the perimeter of the shape, the greater the AoB, since AoB is essentially the thickness of the perimeter/circumference. *(Ref. Annexure A for proof) Similarly, it is easy to observe that AoT is a direct function of the area of the shape of the Pizza.** (Ref. Annexure B for proof) Assuming the perimeter/circumference thickness equal across pizzas, whatever the shape, we have the following theorem:

Efficiencies of the shapes of two pizza can be compared by comparing the ratios of the area of the shape to the perimeter of the shape where,

Pizza Efficiency Factor, § = AoT/AoB ~ Area of Shape / Perimeter of Shape. Therefore, the greater the § for a Pizza the greater its efficiency

Phew!

A break here would be advisable before reading the rest. Or, you may just go back to whatever you were doing before.

Anyway, to define our problem better in this context:

Does a square pizza have a greater § than a circular pizza?

Yet, we see that the problem isn’t complete. We need to also know which square pizza to compare to which circular pizza, right? Should we assume the size of the box constant? Should we assume area of the pizzas equal and then compare? We can’t possibly go further before defining our specific case. So here are three.

Case 1
The Square Pizza Base Is Made By Chopping Off The Excess Parts From The Circular Pizza Base.
Square Inscribed In The Circle

Diagonal of the Square
√2 s = 2r

Pizza Efficiency Factor (Square)
§S=s2/4s = 2r2 /4√2 r = √2 r/4 = 0.354 r

Pizza Efficiency Factor (Circle)
§C = πr2 / 2πr = r/2 = 0.5 r

§S < §C

Therefore, If a square pizza base were to be made by chopping off the excess from the circular pizza base, the circular pizza would be more efficient. Than the square pizza.

Case 2
The Two Pizzas Have The Same Amount Of Toppings.
Areas Equal.

Area of the Square
s2 = πr2

Thus, Side of the Square
s = √π r

PizzaEfficiencyFactor(Square)
§S = s2/4s = πr2 /4√πr = √πr/4 = 0.443 r

Pizza Efficiency Factor (Circle)
§C = πr2 / 2πr = r/2 = 0.5 r

§S < §C

Which implies, If the amount of toppings on the pizza were equal, A circular pizza is more efficient than a square pizza!

Case 3
The Two Pizzas Come In The Same Box.
Circle Inscribed In The Square.

Side of the Square
s = 2r

Pizza Efficiency Factor (Square)
§S = s2 / 4s = 4r2 / 8r = r/2 = 0.5 r

Pizza Efficiency Factor (Circle)
§C = πr2 / 2πr = r/2 = 0.5 r

§S = §C

Which means, If a square and a circular pizzas were to come in the same box, they are both equally efficient pizzas!

Ok, wow!
Conclusion:

The circular pizza shape simply makes better sense. It is not only a because of the old toss n’ twirl but also because of its general efficiency.

However, if the box were to be assumed constant as in Case 3, it makes no difference to the efficiency whether the pizza is circular or square. But with the square pizza, the AoT is greater, which means more of the good stuff.

If we are to compare pizzas of the same price and make the assumption that price of the pizza is a direct function of its AoT, then Case 2 must be chosen as our case. In which case, a circular pizza is more efficient, despite having the same amount of toppings as the corresponding square pizza.

Ordering pizza is evidently a major design decision around here.

Recommendations:
Smarter Pizza (Assuming Case 3): Square Pizza, Andheri East
Smarter Truck Booking: Trip In.

………………………..

*There is no Annexure A. There never was.
** There is no Annexure B either. Tough Luck.

Written by Anand Nair
Artworks by Jyothi Iyer

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