PerpetualImperfexion
April 20th, 2012, 11:29 PM
Ok this is going to get a bit confusing. This is an attempt at theoretical math. For whatever reason though, my figures aren't matching.
Well before I get into the 3-dimentional figures (pyramids) lets take a look at what make me think this way.
We all know that 1/2*b*h = A. Why does this work? Well there are two reasons that I've seen. The first one I think is the one they teach in the books. If you draw a rectangle with a width = to the base of the triangle and a length = to the height of the triangle, then you draw the triangle in there it'll be pretty obvious that the trianlge is = to half of the square, and you can prove it by figuring the area of the other two triangles, adding them together, and comparing them to the triangle. The other way (this is where it gets confusing) is to take the average of the base and the "peak" of the triangle. The peak is of course always 0. So in other words you will just divide the base by 2. After that you multiply this by the height. Why would taking the average of the base and the peak make sense? Well if you were to take a triangle and put it on graph paper, take the length of all the horizontal lines you would get the area. But instead of adding them together couldn't you take their average and multiply it by the amount of lines (the height). Further more instead of taking the average of all those lines, why not take the average of the first and last "lines". The peak is considered a line with a length of zero, the base is the other line. After you get their average you multiply it by the height. Both of these are methods to find the area of a triangle. I'm gonna focus on the second method.
So what does this "second method" have anything to do with the volume of a pyramid? Well there are tons of different types of pyramids. In this example I'll talk about a 5 sided pyramid with a rectangular base. First I'm gonna take the area of the rectangular base. Now it is agreed that a pyramid is just a series of shapes that are stacked up on each other. These shapes are all equal in proportion. As the pyramid goes up in levels not only are the proportions equal but the amount they change by in each level is also equal. So in other words the "peak" of the pyramid is equal in proportion to the base. The peak has an area of 0. So technically if you take the area of the base and average it with the area of the peak (zero) you will get the area of the shape that is in the middle. Now if you take this value and subtract it from the area of the base and add it the area of the peak you will end up with a prism of some sort. If you are using a rectangular base you will end up with rectangular prism. When finding the area of a rectangular prism you simply take length*width*height. I would like to point out that through all these steps the height never changes. By this point I've probably lost you...
My point is that if you take the steps I took to find the volume of a 5 sided pyramid you don't get the same number you get when you use the "proven formula" which is 1/3*bl*bw*h (bl = base length and bw = base width)
Some examples:
bw: 2
bl: 2
h: 2
My volume = 4 cubed units
Proven volume = 2.6666... cu
bw: 3
bl: 4
h: 3
My volume = 18 cu
Proven volume = 12 cu
bw: 4
bl: 5
h: 3
My volume = 30
Proven volume = 20
As you can see my method for whatever reason gives a value that is 50% more of the actual answer. My method makes sense (at least to me) so I don't understand why it is giving a false answer. I assume because the value it is off by is always the same I'm forgetting a step. I see no reason to multiply my answer by 2/3. So what's wrong with my theory?
Well before I get into the 3-dimentional figures (pyramids) lets take a look at what make me think this way.
We all know that 1/2*b*h = A. Why does this work? Well there are two reasons that I've seen. The first one I think is the one they teach in the books. If you draw a rectangle with a width = to the base of the triangle and a length = to the height of the triangle, then you draw the triangle in there it'll be pretty obvious that the trianlge is = to half of the square, and you can prove it by figuring the area of the other two triangles, adding them together, and comparing them to the triangle. The other way (this is where it gets confusing) is to take the average of the base and the "peak" of the triangle. The peak is of course always 0. So in other words you will just divide the base by 2. After that you multiply this by the height. Why would taking the average of the base and the peak make sense? Well if you were to take a triangle and put it on graph paper, take the length of all the horizontal lines you would get the area. But instead of adding them together couldn't you take their average and multiply it by the amount of lines (the height). Further more instead of taking the average of all those lines, why not take the average of the first and last "lines". The peak is considered a line with a length of zero, the base is the other line. After you get their average you multiply it by the height. Both of these are methods to find the area of a triangle. I'm gonna focus on the second method.
So what does this "second method" have anything to do with the volume of a pyramid? Well there are tons of different types of pyramids. In this example I'll talk about a 5 sided pyramid with a rectangular base. First I'm gonna take the area of the rectangular base. Now it is agreed that a pyramid is just a series of shapes that are stacked up on each other. These shapes are all equal in proportion. As the pyramid goes up in levels not only are the proportions equal but the amount they change by in each level is also equal. So in other words the "peak" of the pyramid is equal in proportion to the base. The peak has an area of 0. So technically if you take the area of the base and average it with the area of the peak (zero) you will get the area of the shape that is in the middle. Now if you take this value and subtract it from the area of the base and add it the area of the peak you will end up with a prism of some sort. If you are using a rectangular base you will end up with rectangular prism. When finding the area of a rectangular prism you simply take length*width*height. I would like to point out that through all these steps the height never changes. By this point I've probably lost you...
My point is that if you take the steps I took to find the volume of a 5 sided pyramid you don't get the same number you get when you use the "proven formula" which is 1/3*bl*bw*h (bl = base length and bw = base width)
Some examples:
bw: 2
bl: 2
h: 2
My volume = 4 cubed units
Proven volume = 2.6666... cu
bw: 3
bl: 4
h: 3
My volume = 18 cu
Proven volume = 12 cu
bw: 4
bl: 5
h: 3
My volume = 30
Proven volume = 20
As you can see my method for whatever reason gives a value that is 50% more of the actual answer. My method makes sense (at least to me) so I don't understand why it is giving a false answer. I assume because the value it is off by is always the same I'm forgetting a step. I see no reason to multiply my answer by 2/3. So what's wrong with my theory?