How to Make a Circle Cake from a Square
Cake Baking Mathematics
How to "square the circle" or alternatively "circle the square" is a classical geometry problem about constructing a perfect circle from a perfect square (and vice versa) using only a finite number of steps and only a compass and straight edge. In 1882, the mathematician Ferdinand von Lindemann definitively proved that the construction was not possible under the constraints.
In the case of making circular cakes from square pans, such restrictions are not necessary, nor is any high degree of precision needed. With a ruler and protractor, you can dissect a square into several pieces which can then be rearranged to form a regular 12-sided polygon (dodecagon) that closely resembles a circle. When slathered with thick frosting, it will be practically indistinguishable from a real circle. This is a tasty practical application of the classic polygonal dissection problem.
Step 0: Preparing the Cake
In this dissection of a square into a "circle-ish" shape, some parts of the cake's edge will end up in the middle, while some of the middle will end up on the outer edge. Since cakes tend to be higher in the center, you should first horizontally slice off a little cake from the peak to level everything out. This will make it look neater in the end and make it easier to frost.
Step 1: Measuring and Calculating
Measure the length of the square cake and divide this number by 3.346065 and round the answer to the nearest millimeter or 1/16 of an inch. The number you get will be the side length of the resulting dodecagon. Call this number D.
For example, if the square cake is 9 inches along the side, compute 9 ÷ 3.346065 = 2.689726, which means D is about 2 and 11/16 inches. (To figure out 16ths of an inch, multiply the decimal part of your answer by 16 and round to the nearest whole number. In this case, 0.689726 times 16 approximately equals 11.)
Step 2: Making the Initial Cuts
Starting at the four corners of the square cake, cut inward at an angle of 45 degrees, making cuts of length D ( the number you calculated in Step 1). See image below. If the cake's sides are 9 inches, then the lengths of these four cuts are 2 and 11/16 inches.
Step 3: Making Angles of 150 Degrees
Using a protractor, cut eight new line segments, each of length D, coming out at a 150-degree angle from the cuts you made in Step 2. The endpoints of these eight cuts should match up so that you have a four-pointed star shape in the center. See image below.
Step 4: One More Cut
The last cut you need to make is on the four-pointed star in the center of the cake. Pick two of the adjacent dented-in corners of the star and connect them with a straight line cut, forming a small equilateral triangle. Your square cake will now be cut into six pieces. See image below.
Step 5: Arranging the Pieces to Make a Dodecagon
Arrange the six pieces of cake as shown in the figure below to make a dodecagon (12-sided polygon). This is a fairly convincing approximation of a circle that will look rounder when frosted.
Some Mathematical Background
The number 3.346065 which we used in Step 1 to calculate D is an approximation of the number (3 + sqrt(3))/sqrt(2). Why do we use this particular number?
If the side length of the square cake is L, then the diagonal is equal to sqrt(2)*L. Looking at the dissection of the square in the pictures above, we can see all of the line segments have the same length D. In terms of D, the length of the diagonal of the square is
D + D*sqrt(3)/2 + D + D*sqrt(3)/2 + D = (3 + sqrt(3))*D
Since this equals sqrt(2)*L, we have D = L ÷ [(3 + sqrt(3))/sqrt(2)], or in simpler terms, D ≈ L ÷ 3.346065.
This dissection of the square into a dodecagon, or equivalently of a dodecagon into a square, was devised by Harry Lindgren in 1951. Lindgren was a British-born Australian engineer, linguist, and mathematician who came up with many other clever dissections of polygons. The square-to-dodecagon dissection is one of the simplest in terms of the total number of pieces (six). Another elegant dissection, often credited to Henry Dudeny, is that of an equilateral triangle into a square. For more about dissection puzzles and partitions of squares, see Tangrams, Egg of Columbus, Ostomachion and Other Dissection Puzzles