Will square pizza cover more percentage of area in box with circular base than circular pizza in box with square base?

Question

Usually, pizzas are circular and they come in boxes with a square base. The circular pizza covers some percentage of area of the square base of the box. Let's say someone makes a square pizza and puts it in a box with a circular base. Will the square pizza cover more percentage of area in the box with circular base than the circular pizza in the box with square base ?

Assume that both the pizzas fit perfectly inside their respective boxes.

Answer 1

A circle in a square consumes more percentage of the area of the square ie., only 21.46% of the area is wasted.
On the other hand Square in circle wastes more area ie., 36.33%

For the circle in the square case to find the area that is wasted by the circle in the square in percentage we can use the following method

((s^2-πr^2)/s^2)*100
=(1 - (πr^2/s^2)) *100 ------ 1
but here
r=s/2
s=2r
s^2=4r^2

so 1 becomes

(1-(π/4))*100
=21.46%

Similarly in the case of the square inside the circle we end up with the equation

(1 - (s^2/πr^2))*100 ------ 2
but here
2r=(√2)s
or
s^2=2r^2
so 2 becomes

(1 - (2/π))*100
=36.33%

So we can conclude that the circular pizza consumes more percentage area of the square box it is in than the vice versa case.