Imp 2 Pow 8 Just Count the Pegs

POW 8

Problem Statement-

For this POW, our task was to find the best formula for finding the area of any polygon that is formed on a geoboard. In order to do this, there are two formulas given to help you. One tells how to get the area of a polygon based on the number of pegs on the boundary. This works as an In-Out table, where In is the amount of pegs on the boundary, and Out is the area. The other formula tells how to get the area by having a polygon with exactly four pegs on the boundary. All you do is tell her how many pegs are on the inside. Using her formula, she can give you the area immediately. Using these two formulas for reference, we have to find a formula that anyone can use to find the area of any polygon by knowing the number of pegs on the boundary and interior.

Process-

The first two equations by Freddie and Sally were a preparation for the final, building up the information in order to find the "superformula".

For Freddie's equation, I drew an In-Out table, with the number of pegs on the boundary for the In/x value, and the area of the polygon for the Out/y value. With this table set up, I made some different polygons on geo-paper (attached), and plugged them into the table. I found the area for all of them by looking at the figure. However, just looking at the figure was not very accurate, so I looked for a pattern between the in and the out, and quickly found a formula that made sense, and I worked with it until I got a formula. This formula was x/2-1=y. This works in all shapes with no interior pegs, as the problem said. (In-Out table attached)

For Sally, I saw that another In-Out table would work the best, and drew one with the same guidelines. With this set up, I made some polygons on geo-paper (attached). I found that the In/x value is the number of pegs on the interior, and the Out/y value is the area. I then filled the table with a few values from the geo-paper, and saw a pattern that soon led to a formula. This formula worked for all values that I plugged in. The formula was x+1=y. (In-Out table attached)

Finding Frashy's equation was extremely difficult, and required a long thought process. I knew that Frashy's equation would have to be a combination of the two, and it would have to incorporate all the data that I found from both equations above. After much thought and much time looking at the two equations, I realized there would have to be two variables. This is because it has to include the variable from Freddie's, and the variable from Sally's. We know that these are the number of pegs on the boundary and the number of pegs on the interior. This means that there are two In values. And, of course, incorporating these variables with each other will give me my Out value. Once again, I went to an In-Out table to organize the information. This table helps a lot for finding patterns. For this table, I had three columns, one for the pegs on the interior, one for the pegs on the boundary, and the area. With this table set