(If you had a round cake, and also wanted to cut it right into n-equal sized pieces, the answer is nice simple; girlfriend can cut each piece into sectors through the suitable same angle. Easy.)

What happens, however, if you have actually a square cake, choose this delicious looking fruit cake?

How carry out you carve up this cake into equal size ‘sectors’ (so that each reduced goes come the center; we’ll watch why this is vital later)

As one example, stop say we have a square cake v each side of 6 inches, how can we reduced this into three same pieces?

How did we create these cuts? Well, the perimeter that the cake is 24”, and also there are three pieces, so each piece will obtain 24/3=8” that perimeter. We start making our first cut come the center, climate count about the perimeter 8”, make the following cut, then repeat. The as straightforward as that; division the perimeter, make sure each piece has the same direct length, and also cut come the center.

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It doesn"t matter if the perimeter goes around a corner, as long as there is contiguous 8" of perimeter for each piece. Since of this, there room an infinite number of continuous solutions depending upon where there very first cut is made.

A an effect of each piece having the very same linear size of perimeter is, if the cake has actually icing approximately the political parties edges, that each item receives the same quantity of this icing. (If, instead, we reduced the cake into three same strips using 2 vertical directly cuts, the center piece would be plunder of sheet icing). Now you have the right to see why it"s vital we cut to the center!
It does not issue how plenty of pieces we need to generate. It"s the very same strategy. Right here is exactly how the cake would certainly look if we had to reduced it into 5 equal pieces. Here, the perimeter of each edge is 24/5=4.8”

Recalling back our geometry native school, the area that the triangle is fifty percent the base multiplied by the upright height. In the chart below, each of these triangles has the same area due to the fact that they each have actually the very same altitude and also base.
Imagine carving up the cake into lots the triangles, each v the exact same base (same distance around the perimeter). Together you deserve to see in the diagram below, each of the triangle AD has actually the exact same altitude (their perpendicular distance from the edge to a line with the facility is the same), and also they have actually the very same base. Each of the triangles AD has actually the very same area.
Similarly, you can see the triangles A and Z have the same area. In fact, as the square is a continual polygon, any triangle with with the very same base top top the edge, and a vertex at the center, will have the exact same area.
It go not even matter if a triangle "wraps" approximately a corner. Together the altitude is the exact same for both partial triangles, the area is just proportional to the direct edge length.
If you understand a sophisticated baker who have the right to make regular n-sided polygonal cakes, this strategy still holds. In a continuous polygon, the perpendicular street to every of the edge from the centroid is constant.

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(Taken come the border you can see how this reduces to the very same strategy that we use for a round cake, and how the inner angles room the same.)
If you have the right to imagine scaling up and also down the cake in size (based top top the origin at the centroid), you deserve to see the this technique is agnostic as to the thickness that the icing (providing the icing is the uniform thickness all the means around). Every piece reduced should obtain 1/nth share of the cake and side icing. (Subtract the difference between a cake with, and also without, icing and also the difference is simply the icing).