# Some geometric properties of the Western Canadian party-cut pizza

Introduction

‘Regina-style’ pizza is characteristic to pizza restaurants in the city of Regina, Saskatchewan, and Western Canada generally.  The Regina-style differs from its Chicago cousin primarily in terms of crust depth, and the style of pizza-cut.  Regina-style pizzas have a medium crust height, and are usually cooked on standard one-inch height pans.  Regina-style pizzas are thicker than most other pizzas found in North America; they are usually topped with generous layers of standard pizza meats and (to a lesser extent) vegetables.  Their thickness makes true ‘pie’ shaped pieces impractical, so Regina-style pizzas are usually cut ‘party’ style, most often in two perpendicular sets of parallel and equidistant cuts (Figure 1).  On a standard 6-cut pizza, this leaves 4 ‘corner’ pieces, 8 ‘edge’ pieces and 4 ‘inner’ (crust-less) pieces.  Extra large pizzas may have one more cut per set (for a total of 25 pieces).

Figure 1. A party cut pizza

To date, the author is aware of no formal treatment of the geometric properties of the party-cut pizza.  The following is a brief description of these properties without proofs, but observations will be studied inductively over the next 40 years by eating a large number of Regina-style pizzas.

Some geometric properties

Assuming symmetry about the centre cuts, all party-cut pizzas can only have a number of cuts equal to any positive, non-zero and even natural number.  Much of the geometry of a pizza can be known simply from the number of cuts.  The formula for determining the total number of pieces for a Regina-style party-cut pizza, p, as a function of the number of cuts, c, is

The number of corner pieces is constant, 4.  The number of edge pieces (excluding corners), e, is determined by

Naturally this reduces to

Therefore, the total number of pieces on the edges of a party-cut pizza is always equal to 2c.  The number of inner pieces can be calculated by subtracting 2c from p.

The area of each piece (of all three types) of a Regina-style party-cut pizza increases as a function of diameter of the pizza for a given number of cuts, but decreases proportional to the number of cuts.  Because the shapes of the piece types are different, a different formula must be used to determine the area of each type of pizza piece.

The area of each inner piece, can be found with the diameter, d,  and the number of cuts:

Calculating the area of the edge (non-corner) pieces requires more information than simply the number of cuts and the diameter of the pizza.  One way of calculating the precise area of each edge piece involves adding the area of a circular segment (A) to the area of a quadrilateral polygon (B) (Figure 2).  Calculating area this way requires some extra measurement, however.

Figure 2. Add quadrilateral polygon to circular segment

It is also possible to calculate the area of edge pieces using integration and the measurement of an angle, θ (Figure 3).

Figure 3. Calculating areas of edge pieces

Finally, the area of the corner pieces is equal to the difference between the total pizza area and the sum of the area of inner and edge pieces.

Conclusion

The Regina-style party-cut pizza is a unique Western Canadian treat with generous toppings.  The piece-specific geometry is more complex than the traditional pie-cut pizza, but the toppings are no less tasty.  In a future post, I will estimate the effect of pizza eating on life expectancy…;)