Ancient Problems in Modern Ways - A Reflection

For our presentation, Caris, Brandon and I tackled the volume of a truncated pyramid. To solve it using modern mathematics, the truncated pyramid was presented as the summation of 3 different shaped, a cuboid, four corner pieces (which formed a pyramid when combined), and four triangular prisms (see picture). I was having immense difficulty condensing the formula after adding the different volumes together and I was not able to come up with the correct formula for the volume of a truncated pyramid when collecting like terms. I couldn't see what I was doing wrong and after discussing with my group mates, Brandon was initially able to solve it by using the ratio of b and a to the ratio of the height of the pyramid H to the height of the truncated pyramid h. The entire method requires solving it through a limit process by using integrals but I was really confused why I wasn't able to just add up the different volumes and combine like terms to get the correct formula. I was sitting there racking my brain over the difficulty and I couldn't help but admire the simplicity of the method of ancient Egyptians. As I discussed in class, all they did was average the areas between the base and the top and multiply it by h. Then, they realized they didn't do it properly so added a "median" area of ab and took the average of all 3 and multiplied that whole result by h. 

I recall a unit sometime either in elementary school or high school where we learned about estimation and I vividly remember sitting there and thinking this has got to be the dumbest unit. Why don't we just do the same amount of work (because educated estimates require some thinking) and actually solve for the correct value. Estimating cannot be overlooked and honestly, it might be the concept we use most in our every day lives out of everything we learned in public school mathematics. It needs to be mentioned that the Egyptians were not regular mathematicians - there is no proof for the logic behind their incredibly accurate mathematics despite the lack of modern mathematics knowledge. But their ability to lean on
"rough", educated estimates is a testament to its applicability considering its success. I went down a YouTube rabbit hole just yesterday watching a stonemason create different shaped stones using a chisel and hammer. His accuracy was remarkable. At one point I thought, "but that isn't a perfect triangular prism" and then it hit me that these stones would still be used to make driveways, stone houses and other things. And if I had seen it in real life as a completed project I would be able to admire its beauty without thinking "wow but this side of the triangle isn't perfect". There's a lot to be said about that. At the end of the day, practically we don't use high-powered lasers to create perfectly level sides when we're shaping stones - we're creating roughly accurate shapes. I don't know, it's something I'll continue to ponder on for sure.