Introduction: Reshape Rectangular Pizza Into Hexagonal Pizza

About: I've built some weird stuff over the years, but most of that stuff has remained unseen by the world outside of me and a few friends. But then one day, one of these friends, he says to me, "Hey Jack, you shoul…

This instructable is more about math than about pizza, although it does involve both.

One of the problems with math problems, is many math problems involve too much abstraction, to the point where the problem seems fake, or at least lacking application in real life.

And truly you will find yourself wondering how this math problem applies to your kitchen, unless you have the exact same sized frozen pizzas, specifically 7.0 inch by 8.0 inch rectangle-shaped pizza, and you also need to make those pizzas fit into a circular pan having a diameter of 9.0 inches.

The numbers seem made up. Seriously, the integers 7, 8, and 9? Just like that?

You probably think I am making those numbers up... but this time I have the pictures to prove it! I have pictures of pizza and pans, right next to a plastic rulers and graph paper!

You see, all the trouble started, when I went to the local megalomart, to buy some cheap frozen pizzas. To my dismay, the manufacturer of the pizzas I used to buy, had changed their format, from circle shaped pizzas, to rectangle shaped pizzas. What was worse, their new rectangle shaped pizza would not fit in my favorite, steel, PTFE-coated, microwave-sorbing, round pizza pan!

One solution to this problem, is to reshape the rectangular pizza into something approximating a circle... like a regular hexagon. That shape will more easily fit in a circle shaped pan. Also a hexagon shaped pizza can be cut into slices that are equilateral triangles (which are also equiangular, 60-60-60), and then cut further into 30-60-90 triangles.

Now if anyone reading this is a foodie, a gourmand, a battle-hardened kitchen veteran... I can imagine you might have some complaints. Prefab frozen pizza!? Cooked in a microwave oven!? What could be more lazy and uninspired than cooking a frozen pizza, in some gimmicky microwave-sorbing pan that probably came with the words, "AS SEEN ON TV" printed on the box. This is not Cooking! This is not Cuisine!

To be fair, I think I gave you foodies some warning at the beginning of this intro, by telling you this 'ible involved mathematics, more-so than the food, the pizza itself.

However, if you stick with me on this one, I think you will find pizza tastes better when it is sliced into 30-60-90 right triangles.


You'll see.

Step 1: A Pizza Problem

The frozen pizzas I have are very close, in shape, to rectangles, with one side having length 7.0 inches, and the other having length 8.0 inches. How much variance, or uncertainty, is there in those measurements? Perhaps as much as 0.2 inches.

The pan I want to use is circular, with an interior diameter close to 9.0 inches (and a radius close to 4.5 inches). How much variance, or uncertainty is there in that measurement?

Well, the walls of the pan taper downward, so the pan is more narrow at the bottom than the top. The very bottom of the pan might be as narrow as 8.9 inches in diameter, and at the very top (of a pan that is about 1.0 inches deep) the diameter might be as wide as 9.3 inches.

In any case, the first picture, with my frozen rectangular pizza resting on top of my circular pan, but not inside it, strongly suggests: This pizza does not fit in this pan!

Step 2: Criteria for Success

I have some goals for this hack. Those goals are presented here in the pictures for this Step, in some classy black-and-white graphics, and also in the numbered list below:

  1. Goal # 1: I want to make only two cuts while the pizza is frozen. Why only two? Because cutting frozen pizza is hard work.

  2. Goal # 2: I want all the pieces of frozen pizza to fit inside the pan.

  3. Goal #3: Make it so all the pieces, resulting from cutting the pizza after it is cooked, are all approximately the same size and shape. Why is this a goal? Because symmetric food is pretty.

Step 3: Cutting Corners

I present a method for reshaping my rectangle shaped pizza.

This method consists of making two cuts, and removing two triangle shaped pieces, as shown in the diagram attached to this step. Moreover the shape for these cut-off corner pieces, is always a right triangle, with angles {30, 60, 90 degrees} also called a 30-60-90 right triangle.

I want to consider a range of sizes for the cut off pieces, and I plan to do this with just one variable, f, which is length of the cut-off triangle, on its shortest side.

Because these are 30-60-90 triangles, the lengths of the other two sides vary in proportion to f. I call them g = (3^0.5)*f, and h=2*f.

where (3^0.5) is another way to write the square root of three, and it is approximately equal to 1.732050808.

Regarding the range of f, f can have values in the range:

fmin= 0 = 0.000 inch, to fmax=4/(3^0.5) ~= 2.309 inch.

The constants in this problem have names too. The rectangle has dimensions, a by b, where a = 7.0 inch, and b = 8.0 inch.

The circle representing the pan, has diameter d = 9.0 inch, and radius r = 4.5 inch.

The pictures for this Step include a drawing of this 7x8 pizza, and a printed paper version of the same, on my kitchen counter, and also cut into pieces in my pizza pan.

For the version shown in the pictures for this Step, f = 1.500 inch.

Step 4: The Prettiest Pizza

I can say, with some confidence, that any f in the range [1.500, 2.309] will work, for these size frozen pizzas (7 inch by 8 inch) I have, and my pan (9 inch diameter).

However, to satisfy Goal 3, I must try to make all the finished pieces close to the same size. Also I want to make a finished pizza that is pretty to look at, because, after all, I have to take pictures of it for this instructable.

To accomplish all this, I increase f to its maximum value,

f = fmax= 4/(3^0.5) ~= 2.309 inch ~= 2 + 5/16 inch.

This Step includes a picture of a 2-dimensional, paper simulation of this pizza, and the same paper pizza cut twice, and assembled in the pan.

Step 5: Kitchen Work - Step a - Grease Pan, and Pre-heat.

It is time to go to work in the kitchen! First thing: my steel, microwave-sorbing pizza pan must be greased, to prevent the pizza from sticking to it.

I grease it using some cooking oil that comes in an aerosol can.

Then I pre-heat this pan, by putting in my microwave oven for 1m+0s, on HIGH power, so it can sorb some waves and get heated up!

Step 6: Kitchen Work - Step B - Make the Cuts.

It is time to make the two cuts, the ones that define the two cut-off triangle pieces.

To help me make this cut accurately, I made a plastic template of the shape I want.

I hold this template against the pizza. Then put my trusty kitchen knife exactly against the edge where the cut should go.

In the next step I explain how I made the template.

Step 7: (Optional Step) - a Plastic Template to Guide the Cuts.

If I only want to do this once, then a plastic, washable template is unnecessary.

After all, I could just print a paper template, and hold that against the pizza.

However, the voices in my head told me to make this. In fact those same voices gave me the idea of cutting it from the flat side of a large, HDPE plastic bottle. This kind of plastic is thin enough to be cut with some sturdy scissors.

I taped some pieces of masking tape onto the plastic, and I used a ruler, and a drawing triangle (another 30-60-90, naturally), and a pencil, to mark the measured lines onto the pieces of tape, before cutting along those lines with the sturdy scissors.

Step 8: Kitchen Work - Step C - Pizza Goes in the Pan. Then Into the Microwave.

The pizza goes in the pan. Then the pan and pizza together, go into the microwave oven. Heat on HIGH for 5m+0s. I use an insulating pad (a.k.a pot holder) to protect my hand from the pan, which is HOT! (Remember, I pre-heated it, in the previous step.)

Step 9: Kitchen Work - Step D - Fresh Garlic and Basil.

While the pizza is busy sorbing waves, it is time to do some chopping!

You can add whatever you want to your pizza.

I am thinking mine could be much improved by adding some chopped, fresh, garlic and basil. So the pictures for this Step show my cutting board, and my trusty kitchen knife, and some little piles of chopped garlic and basil.

Step 10: Step E - Sprinkle Herbs Onto Pizza.

I use an insulating pad, to protect my hand, when I take the pan out of the microwave. The reason why is because that pan gets HOT! And I don't want to burn myself.

I use my knife and plastic spatula, much in the same way as a broom and dustpan. I scoop up those little piles of chopped garlic and basil, and carefully sprinkle these onto the hot pizza.

Step 11: Step F - Back to the Micro.

Again using the insulating pad to pick up the pan, I put it back in the microwave, for its final pump. This time for only 1m+0s.

Step 12: Kitchen Work - Step G - Cut and Arrange.

The climax of this weird journey has finally arrived!

I take the pan out of the microwave. Then I carefully maneuver the pizza from the pan onto the cutting board.

First I cut the big region into, approximately, four equilateral triangles, plus two 30-60-90 triangles.

Of course each 30-60-90 is just what you get when cutting an equilateral triangle neatly in half.

Because I want all the pieces to be the same, I cut those four equilateral triangles in half, into pairs of 30-60-90s.

Then I arrange all the triangle slices into the same pattern I imagined in Goal #3, in Step 2.

I will include that picture in this Step also. So you can look at both, and judge for yourself how close I came to this imagined pretty pizza pattern.

Step 13: Put It on a Plate.

Well, with all this thinking about geometry, I have worked up an appetite.

It is fortunate I have this cooked pizza here in front of me. I am probably going to eat the whole thing myself.

But before I do, I will take just a few more pictures.

On the plate, you can see two 30-60-90 triangles, each the mirror image of the other.

Now, isn't that pretty?


Those beauties aren't going to last for long though. It's a good thing I took a picture of 'em.

Step 14: A Lucky Coincidence.

This hexagon is not a regular hexagon. Although it is very close to being that.

Four sides of this hexagon have length h=2*f, and two sides have length (a-f)

In general, h=2*f and (a-f) will NOT be equal to each other, when starting with any given rectangle, with arbitrary side lengths a and b.

It is just a lucky coincidence, that starting with a rectangle with sides, with lengths a = 7.0 inch and b = 8.0 inch , give h and (a-f) that are very close. How close?

f = (b/2)/(3^0.5)

h = 2*f = 2*4/(3^0.5) ~= 4.6188 ~= 4 + (10/16)

(a-f) = 7-(b/2)/(3^0.5) ~= 4.6906 ~= 4 + (11/16)

In this case, h and (a-f), the different side lengths, differ by about 1/16 inch.

Or maybe it is not a coincidence? Perhaps the pizza manufacturer did this intentionally. By design.

Who knows what wonders are possible in this modern World?


Step 15: Rectangle From Regular Hexagon

By the way, if you were to start with a regular hexagon, and then rearrange it into a rectangle, the ratio (a/b) would be exactly:

(a/b) = (3^0.5)/2 = cos(30 deg) = cos(pi/6 rad) ~= 0.86603

Which is close to (7/8) = 0.87500

The math for this is shown in the picture for this Step, which is a rectangle with sides:

a = 4*(3^0.5)=6.928

b = 4*2 = 8.000

The cut-off triangle has side lengths:

f = 4/(3^0.5)=2.309

g = (3^0.5)*f = 4.000

h = 2*f = 2*4/(3^0.5) = (a-f) = 4*(3^0.5)-4/(3^0.5) = 4.619

And in this case, all the sides of the hexagon, have length h = 2*f.

Pizza Speed Challenge 2020

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