A Pizza for Eight Is 13.86 Inches, Not Sixteen

A twelve inch pizza feeds six people. How wide does it have to be to feed eight?

A third more mouths sounds like a third more pizza, which sounds like sixteen inches. It is 13.86, and a sixteen inch pizza would feed almost eleven people. The gap between those two answers is a whole extra meal, ordered by accident.

Servings follow the area

Nobody eats a diameter. What is on the table is area, and the area of a disc goes as the square of its width:

A twelve inch pizza carries $36\pi$ square inches, so feeding six means $6\pi$ square inches each. That per-person figure is the only thing being held fixed, and everything else follows from it.

Two routes, both short

The ratio route needs no absolute quantity at all. Feeding eight instead of six needs $8/6 = 4/3$ of the area, so the diameter is multiplied by $\sqrt{4/3} = 2/\sqrt3$:

The answer is exact. The 13.86 on a menu is a rounding of $8\sqrt3$, which is pleasant enough to be worth noticing, and it comes from the surd rationalising: $\sqrt{4/3} = 2\sqrt3/3$.

The absolute route arrives at the same place and settles a question the ratio route leaves open. Eight people at $6\pi$ square inches each need $48\pi$, and a disc of that area has $d = \sqrt{4 \cdot 48} = \sqrt{192} = 8\sqrt3$. Working in absolute units makes it clear that the number is a diameter and not a radius, which is worth checking whenever a circle problem is stated in the way pizza is actually sold.

What sixteen inches actually buys

The reflex answer is not random, and it is worth pricing rather than dismissing. A sixteen inch pizza has

Almost eleven, not eight. The error runs in the generous direction, which is why the mistake rarely gets caught at the table. It gets caught when the bill arrives.

The underlying trouble is that $\sqrt k < k$ for every $k > 1$, so applying an area ratio to a length always over-orders. Fifteen percent more width is thirty-three percent more food, and neither figure looks like the other.

The exponent is the thing to find

Once servings are known to follow the second power of a length, everything else is substitution. To multiply the servings by $k$, widen by $\sqrt k$:

Feeding half again as many wants 22.5 percent more width. Doubling wants 41.4 percent. Tripling wants 73.2 percent. Doubling the width, which is what "a pizza twice the size" usually means out loud, feeds four times as many.

The same substitution works for any exponent, and getting the exponent right is the whole job. Ice cream comes in balls, so its quantity follows the cube of a length: a scoop with twice the ice cream is only 26 percent wider, since $2^{1/3} = 1.26$. Paint follows the square, like pizza. A ball of dough of fixed thickness rolled out to feed twice as many has to be twice the weight and only 41 percent wider, so the same order looks generous in one unit and modest in the other.

The question asked the other way round

Inverting equation (4) is often the more useful direction, because a menu gives you diameters and you want head counts:

Ten inches feeds 4.2 people. Fourteen feeds 8.2. Eighteen feeds 13.5. The nonlinearity is easy to underestimate at the point of ordering: the step from 12 to 14 inches adds 2.2 servings, while the step from 16 to 18 adds 2.8, so the same two inches are worth more the larger the pizza already is.

The crust makes the same point from a different direction. Its length grows like $\pi d$ while the food grows like $d^2$, so crust per person falls like $1/d$. A twelve inch pizza carries 6.28 inches of edge per person and a $13.86$ inch pizza carries 5.44. Whether that is an improvement depends entirely on your view of crust, which is outside the scope of the algebra.

Two assumptions that would move the answer

Equation (2) assumes servings are exactly proportional to area, and a real pizza has a crust. Suppose the outer inch is bare, so what feeds you is the disc of diameter $d - 2$. Then the condition becomes $(d-2)^2 = \tfrac43 \cdot 10^2$, giving

Slightly less than 13.86, and the reason is worth a sentence: a wider pizza spends a smaller fraction of itself on crust, so it is proportionally more generous. The topped fraction rises from 69.4 percent at twelve inches to 73.2 percent at 13.86. The reflex answer of sixteen inches is still wrong by miles under either convention.

The other assumption is uniform thickness. Restaurant pizzas of different diameters are rolled to about the same thickness, which is what makes area the right measure. If thickness scaled with width instead, quantity would follow $d^3$ and the answer would be $12 \cdot (4/3)^{1/3} = 13.21$ inches. The exponent is a modelling decision, not a fact about circles, and it is the only place this problem can genuinely be argued about.

Sources and further reading

• Equation (1) and its history — Area of a circle
• The scaling rule in the definition — Similarity (geometry)
• Why exponents decide everything here — Square-cube law

The 13.8564 was checked without ever evaluating equation (1): four million random points were thrown at each disc to measure the area ratio, a grid quadrature confirmed it to six decimals, and a bisection on those measured areas returned 13.8564 inches without any square root being written down.