The diameter of a pizza is not its size
The Sunday Nation
Nairobi,
19 August 2012
Charity Muli is concerned about the price she pays for pizza. She writes: “My favourite pizza restaurant offers three sizes – small (19cm), medium (23cm) and large (30cm). The small one costs Sh390, the medium is Sh650 and the large is Sh880. I worked out the price per centimetre and found Sh25 for the small one, Sh28 for the medium and Sh29 for the large.
“This is the opposite of what I expected – the unit price of the large should be the lowest. Can you explain why the pricing appears to be different from normal business practice?”
Charity; your calculations and expectations are correct but your conclusion is not! It is true that normally, businesses give better price per unit when one buys the larger size of a product. But what exactly do we mean by “size” in the case of a pizza?
Pizzas usually come in the form of a circular disc and the dimensions given by the restaurant are the diameters. The “size” in this case is not simply the diameter. It is the surface area.
Even though the bigger the diameter, the bigger the area; the two quantities are not directly proportional to one another. Suppose for example, that the pizza was square in shape with sides measuring 10cm each. The area would be 100 square centimetres (10cm x 10cm). What if the lengths were doubled to 20cm each; would the area also double? Of course not: 20cm x 20cm is 400sq.cm – four times that of the first one!
In the case of a circle, the area is given by the square of the radius (distance from the centre to the edge) multiplied by pi (3.14). But the dimensions given are diameters; that is the total width of the pizza. This is double the radius thus to get the corresponding area, we must first divide the figures by two; then square the result; and finally multiply by pi.
It turns out that the small pizza is 283 square centimetres, the medium is 415sq.cm and the large is 707sq.cm. We now use these areas to compare the prices of the pizzas per unit.
These come to about Sh1.40 per sq.cm for the small size, Sh1.60 per sq.cm for the medium and Sh1.25 per sq.cm for the large one. It still doesn’t follow the “normal business practice” perfectly. The medium size turns out to be the most expensive; but, at least, the largest one is also the cheapest to buy.
That, however, shouldn’t be cause for concern. We should not rush to complain that the restaurant is unfair. The way I see it is that this is a result of market dynamics – the effect of demand on prices. I suspect that the demand for medium pizzas is far greater than that of the other sizes. Thus the restaurant has seen an opportunity to recover some of the discount given on the large one. It’s nothing personal; it’s just business!
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