Abstract Hypothetically if you divide a disk equal between an even amount people is not as easy as it seems. For instance, using a regular utensil making cuts that are straight, there will be one point where all the cuts hits the middle. Merely, there are other ways to use the Pizza Theorem which also associates with elementary geometry and also integral calculus. This paper deals with the so called “pizza theorem”.BackgroundDo you ever feel like you are getting the rough end of your pizza pie when you hang out with your friends? Fortunately, I will tell you how to make sure that you get an even amount of pizza every time using the Pizza Theorem. The theorem is, if a disk or a circular pizza is divided into 8, 12, 16 or any multiple of 4, slices by making cuts at equal angles from a specific point, then the sums of the areas of alternate slices are equal. However, the Pizza Theorem will not work if the disk or pizza is divided into only 4 slices nor will it work if they’re an odd amount of slices. 2 The pizza theorem was first shown to prove a math problem by L.J Upton in a mathematics magazine. In the magazine, Upton was asking how someone can evenly split a disk between two people if the disk’s center was misinterpreted and sliced up unevenly. Upton didn’t ask for two cuts; he asked readers to prove the theorem in the case of four cuts. The answer to this problem was solved by Michael Goldberg. The way he solved it was by a direct manipulation of the algebraic expressions for the areas of the sectors. 1 Other people, Carter and Wagon, which have also solved this question provide an easier and different method of answering. This method shows how they solved the proof without words. They stated if a circle is divided into eight slices by making eight cuts at 45 degrees from the same point, then the sums of the areas of alternate slices are the same. You can also see that each pair of numbers in figure one appear in both the blue and yellow sections. Therefore, this proves that the blue areas are congruent to the yellow areas. Figure one also shows how to divide the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent piece in an even-numbered sector, and vice versa.3 This proof by picture in figure one works no matter where the center is, Based on figure One this is how you prove the pizza theorem without words. You can also solve the Pizza Theorem with algebra such as how Michael Goldberg solved it. The idea of choosing the Pizza theorem to study was for a couple of reasons. The first reason why i chose this theorem is because it can be applied in the real world and also solve problems. One way it was used was by Stan Wagon which who carved a piece of granite using this theorem and it won him an award. In the article the Invisible Handshake its states “He had been interested in my mathematical sculpture for some time, and we collaborated on a small granite piece illustrating a dissection proof of the pizza theorem; it is used as the award for a collegiate competition in Minnesota.” This shows how it can be used in the real world because he used the theorem to divide granite. This can also solve problems such as figuring out how to give an equal amount of pizza to you and your friend if they are cut in different and sizes. Another reason why is because everyone including me loves pizza and i also enjoy math, therefor when i saw pizza and math together i couldn’t deny it. Another reason is that now there won’t be any more arguments on who gets what piece of pizza with friends because you will know that you and your friends are getting a fair amount. ApplicationAs I stated before the Pizza Theorem can be applied to various aspects in the world if you have a shape that would resembles a disk. The proof for the Pizza theorem shows how the Pizza Theorem can be applied. If you allow p to be a point on the inside of the disk, and allow n to be a numeric value that can be divided by four and than and more or equivalent to 8. If you create n sections of the circle with equivalent angles by taking a random line through p, rotating the line n/2 ? 1 times by an angle of 2?/n radians, and cutting the circle on each of the remaining n/2 lines. Number the sections in succession in a forward or reverse way. If you do this then you can see how the Pizza Theorem works.What IfWhat if an odd number party tried to apply the pizza theorem to a pizza pie they ordered. Would it work or would there be a problem. They would run into a problem for many reasons. The pizza theorem can only be applied to an even amount of sectors and have to be multiple of fours, such as four, eight, twelve, and ect. An odd number of slices would not work also because An odd number of sectors is not possible with straight-line cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors.ConclusionFor further research the pizza theorem
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