Video Transcript
In this video, believe it or not,
we’re going to look at some of the mathematics behind slicing and equally sharing
cake.
We’re going to begin by imagining
that you have an iced cake with a circular base. You might have even seen the
nutritional information on these things. Who would want to share one of
these things between 20 people is beyond me. But let’s now imagine you do wish
to share this cake between the recommended 20 people. How do you ensure that each person
has the same amount of cake and the same amount of icing?
Well, that’s quite simple. You measure multiples of 360
divided by 20, which is 18 degrees from the center of the cake. The area of each sector will be
equal. And since volume is just the area
of the cross section multiplied by its depth, we can see that each person will
receive exactly one twentieth of the cake. Similarly, they’ll receive one
twentieth of the icing. And if you had 𝑛 people, well, we
use the same idea. This time, you measure multiples of
360 divided by 𝑛 degrees from the center of the cake. And each person now receives one
𝑛th of the cake.
Next, we’re going to imagine that
we have a cake with a square base. This time, how do we ensure that
each of the 20 people get not only the same amount of cake but also the same amount
of icing? You might want to use a little bit
of trial and error. Perhaps you try to cut the cake
using horizontal and vertical slices to achieve 20 equally sized slices. Well, that’s fine and well. Each slice of cake is indeed the
same size. But the lucky people who get a
slice from the outside of the cake get quite a lot more extra icing than those who
get a slice from the inside.
Next, you might try a similar
technique to the circular cake, measuring multiples of 18 degrees from the center of
the cake. Does this give us the same amount
of icing or cake in each slice? Well, it might look like it, but it
actually doesn’t. Let’s look at a specific example of
a cake measuring 10 centimeters by 10 centimeters.
We can use a process called
disproof by counterexample to show that slicing this cake at the center into angles
of 18 degrees will not yield the same area and therefore volume of cake. We’ll calculate the area of the two
pieces highlighted in this square.
I begin by adding the diagonals of
the square. And we know that these diagonals
meet at an angle of 90 degrees. We also know that the two diagonals
bisect each other perfectly. So, if we call these two lengths 𝑥
centimeters, we can use the Pythagorean theorem to work out their lengths.
Remember, the Pythagorean theorem
says that the sum of the squares of the two shorter sides is equal to the square of
the longer side, or the hypotenuse. In our triangle, the two shorter
sides are 𝑥 centimeters in length and the longer side is 10 centimeters. So we can say that 𝑥 squared plus
𝑥 squared equals 10 squared. 𝑥 squared plus 𝑥 squared is two
𝑥 squared, and 10 squared is 100.
We’re going to solve this equation
for 𝑥. Dividing through by two tells us
that 𝑥 squared is equal to 50. And then finding the square root of
both sides, we obtain 𝑥 to be equal to the square root of 50. Now, there are a couple of points I
need to make here. Usually, when we find the square
root of both sides of the equation, we’re interested in both the positive and
negative square root. But here 𝑥 is a length. So, actually, we’re only interested
in the positive square root of 50.
We can also simplify the square
root of 50 using our knowledge of surds. But since we’re going to be using
the select calculations, it’s really unnecessary. And we obtain this length to be
root 50 centimeters. This might not seem like enough
information. But we know that this angle here is
half of 90 degrees. It’s 45 degrees. And if we subtract 18 and 45 from
180, since that’s the angle sum of a triangle, we obtain this angle to be 117
degrees.
We now have enough information to
be able to use non-right-angle trigonometry to calculate the length of the missing
side. Once we’ve done that, we can then
use the formula for the area of the non-right-angle triangle. We’re going to use two parts of the
law of sines to calculate this missing length that I’ve now called 𝑎.
Using the general form for labeling
a non-right-angle triangle, and I see that angle 𝐴 is 45 degrees, angle 𝐵 is 117
degrees, and side 𝑏 is root 50. So 𝑎 over sin of 45 equals root 50
over sin of 117. I multiplied both sides of this
equation by sin of 45 degrees. And so, 𝑎 is equal to the square
root of 50 over sin of 117 times sin of 45. That’s 5.6116 and so on.
The formula for the area of a
non-right-angle triangle is a half 𝑎𝑏 sin 𝐶. So the area of our triangle is a
half times root 50 times the nonrounded version of 5.6116 times sin of 18, which is
6.13 square centimeters correct to two decimal places.
Let’s repeat this process for the
second slice. Let’s relabel this triangle, with
one side being 5.6116 centimeters and the side we’re trying to find labeled in green
𝑎. We can use angles on a straight
line to see that this angle is 63 degrees and this one is 99 degrees. Substituting our values into the
formula for the law of sines, and we see that 𝑎 over sin 63 equals 5.6116 and so on
over sin of 99 degrees.
Once again, we multiply both sides
by sin of 63, and we find that 𝑎 is equal to 5.062. This time, the area of the triangle
is a half times 5.6116 times 5.062 times sin of 18, which is 4.39 square centimeters
correct to two decimal places.
Clearly, this is significantly less
area than the area of the triangle above. And so assuming the cake is indeed
in the shape of a prism, the volume of the slices of cake are different. And by showing that these two
slices are different in volume, we’ve shown that sharing the cake in this form
cannot yield equally sized pieces. So what else can we try?
Well, the solution is a little
easier than you might think. This time, rather than working from
the inside of the cake, we work from the outside. We divide the perimeter itself into
20 equal strips. In the case of our 10-by-10 cake,
its perimeter is 40 centimeters. So, when we divide that into 20, we
find that each strip is two centimeters in length. We then construct a triangle from
the end of each strip into the center of the cake.
Now, this might look really similar
to what we just did. But let’s calculate the area of
each of the triangles. This time, we can go straight to
the formula area of a triangle is a half times the length of the base times its
perpendicular height. The base of every single triangle
is two centimeters, and the perpendicular height of each triangle is a half of the
total length of the square. That’s 10 divided by two, which is
five centimeters.
So every single triangle has an
area of a half times two times five, which is five square centimeters. And we’ve seen that if the area on
the top of the cake is the same, then the volume and the total amount of cake is
also equal. Of course, since we’ve divided the
perimeter evenly, this means that not only will each person receive the same amount
of cake, they’ll also receive the same amount of icing. And what’s really cool here is that
it doesn’t matter if we don’t start with a corner. For example, we might start like
this. And beginning to share the cake
between 20 people would look as shown.
This triangle still has a base of
two centimeters and a perpendicular height of five centimeters. So its area is still five square
centimeters. And then we calculate the area of
the piece on the corner by splitting it into two further triangles. Each has a base of one centimeter
and once again a perpendicular height of five centimeters. And so the area of the smaller
triangle is a half times five times one, which is two and a half square
centimeters. And there’s two of them. So the area of the piece on the
corner is 2.5 times two, which is five square centimeters, as before.
We can even generalize this and
share a square cake of side length 𝑥 centimeters between 𝑛 people. The total perimeter of this square
is four times 𝑥, which is four 𝑥 centimeters. We’re going to split the perimeter
into 𝑛 equal strips. That means each strip is going to
have a width of four 𝑥 divided by 𝑛 centimeters. The perpendicular height of our
triangle is a half 𝑥. So the area of our triangle is a
half times four 𝑥 over 𝑛 times a half 𝑥. And we can cancel all of these
twos. And we obtain the area of every
single one of the triangles on our cake to be 𝑥 squared over 𝑛 square
centimeters.
For a cake of depth ℎ centimeters,
we know that every single slice will have the same volume. That’s 𝑥 squared over 𝑛 times ℎ
cubic centimeters. Similarly, the icing around the
edge is shared into equal-size rectangles, meaning the amount of icing is the same
too. What a result! And what about a more fancy
cake? Let’s say a cake in the shape of a
regular hexagon. Well, the same strategy still
holds. In a regular polygon, the
perpendicular distance to each of the edges from the center of the polygon, or
centroid, is constant, meaning that the area of each triangle and therefore the
volume of cake will also be equal.
And there’s so much mathematics
behind cutting cake it’s quite ridiculous. There are algorithms to ensure that
cake shared between a number of people without measuring implements is fair, methods
to share a cake with pieces cut out of it, ways to cut a circular cake to ensure it
stays as fresh as possible when stored overnight, and, believe it or not, the
mathematical notion of proof by contradiction is somehow tied to flourless chocolate
cake. Who knew maths could be quite so
tasty?
