# A tale of two pizzas

 Image Credit: Pizza Slice Clip Art from vector.me (by johnny_automatic) My wife and I were in a pizza shop near Warsaw old town. We were looking at the menu and saw that 31cm pizza costs about 30 PLN and a 61cm pizza costs about 70 PLN. There is of course a little variation with the exact toppings. Based on this my wife said that we would be better off getting two 31cm pizzas rather than one 61cm pizza.

So I bought the one 61cm pizza.

But why?
In all fairness, my wife did immediately realise her mistake…

The pizzas, which are assumed to be circular, are specified by their (approximate) diameter $$d$$, which is of course twice the radius $$r$$. Thus surface area of the pizza is given by

$$A = \pi r^{2}$$.

Importantly the area varies as $$r^{2}$$ and not simply as $$r$$. Thus a pizza with twice the diameter, or equivalently twice the radius of a given pizza, has four times the surface area, not twice the surface area.

Simply

$$\frac{A_{2}}{A_{1}} = \frac{\pi (2r)^{2}}{\pi r^{2}} =4$$.

Roughly, given that the pizza was not a perfect circle and that 61 is not quite twice 31, I get about twice as much pizza buying the single 61cm as compared with buying two 31cm pizza.

Furthermore, $$70/60 \approx 1.2$$. So I get near enough twice the pizza for my money by getting one 61cm pizza as compared with two 31cm.

The pizza

The result of my mathematics!

Above is the said 61cm pizza. It was very good.

In conclusion