Pizza, a beloved culinary delight, comes in various sizes. To better understand the implications of pizza size on the amount of pizza consumed, we establish a new standard unit called the standard pizza radius, denoted by the letter \(a\), which measures 6 inches. This article examines how the area of a pizza changes with size and demonstrates that one extra-large pizza can provide more pizza than two standard-sized pizzas.
The area of a pizza, approximated as a circle, with a radius of one standard pizza radius (\(a\)) is given by the formula:
\[ \text{Area}_{\text{standard}} = \pi a^2 \]
For an extra-large pizza with a radius \(r = 1.5a\), the area can be calculated as follows:
\[ \text{Area}_{\text{extra-large}} = \pi (1.5a)^2 = \pi \cdot 1.5^2 \cdot a^2 = \pi \cdot 2.25 \cdot a^2 \]
The combined area of two standard pizzas with radius \(a\) is:
\[ \text{Area}_{\text{two standard}} = 2 \cdot \pi a^2 \]
Comparing this to the area of one extra-large pizza:
\[ \pi \cdot 2.25 \cdot a^2 > 2 \cdot \pi a^2 \]
Simplifying, we find:
\[ 2.25 \pi a^2 > 2 \pi a^2 \]
Thus, the area of one extra-large pizza is greater than the combined area of two standard pizzas.
To determine the minimum radius \(r = n \cdot a\) for the extra-large pizza to always have a greater area than two standard pizzas, we start with the inequality:
\[ \pi (n \cdot a)^2 > 2 \cdot \pi a^2 \]
Simplifying, we get:
\[ n^2 \cdot \pi a^2 > 2 \cdot \pi a^2 \]
\[ n^2 > 2 \]
\[ n > \sqrt{2} \]
\[ n > 1.4142 \]
Therefore, the radius of the extra-large pizza must be at least \(\sqrt{2}\) times the radius of a standard pizza to ensure its area is always greater than that of two standard pizzas.
In Italy, according to the Disciplinare verace pizza napoletana (guidelines for authentic Neapolitan pizza), the radius of a pizza ranges from 22 to 35 cm. Let’s compare the area of two pizzas with a 22 cm radius to one pizza with a 33 cm radius.
\[ 2 \cdot \pi \cdot 22^2 = 2 \cdot \pi \cdot 484 = 2 \cdot 1520.56 = 3039.52 \, \text{cm}^2 \]
\[ \pi \cdot 33^2 = \pi \cdot 1089 = 3419.46 \, \text{cm}^2 \]
This calculation confirms that one pizza with a 33 cm radius has a greater area than two pizzas with a 22 cm radius. Therefore, it is mathematically established that consuming one extra-large pizza results in more pizza than consuming two smaller ones.