I love getting these style of comments! Thank you for sharing your thoughts and I hope the video has helped you in some way to think outside of the box about this prism problem. Cheers and welcome to 1mjourney.com
Hi! In this case do not forget that the frustum can be composed of a bigger pyramid (call it P) minus the top pyramid sliced off (call it p). So a frustum will be P - p, and both P and p have a 1/3 in their formula since they are pyramids. Therefore, the 1/3 can be factored out which is what you need in the frustum case. Let me know if this explanation helps. Welcome to 1mjourney.com
Hi. “w” is width, “d” is the depth, and “h” is the height. One can use any variables they like, but thank you for your comment as it does point out that non-English speaking learners go through the video. Happy learning!
Hi! I am using algebra here to simplify the right hand side by working with the height, width, and depth. I am not sure which term(s) you are referring to, but you can let me know and I will try to follow up. Cheers!
Why don't you use a "c1" factor for the fist 2 pyramids, a "c2" factor for the second pair of pyramids and a "c3" factor por the third pair of pyramids? I don´t know why is always a "c" factor. Thanks!
Great question! The idea and assumption in the proof is that for any pyramid the volume of it is Base * Height * Constant. The base * height people do not have a problem with, but of course it is the Constant that tickles our mind. Note: what you point out is legitimate and you may want to call the Constant "c1", "c2", etc. on the assumption that each constant might be different, but this would imply that different pyramids have different constants and thus volume of pyramids would not have a unique formula hence a contradiction to the assumption that the constant is unique. If you are not convinced then assume the c1, c2, etc are not all equal to c (proof by contradiction), then this would show you that when you add up the overall volume of all the pyramids that are constructed from the Prism would not sum up recreate it. The volume then would be either less or more than the overall Prism - a contradiction. I hope this helps you. Let me know. Welcome to 1mjourney.com
I am disappointed that you used your result from rectangular pyramid to prove the triangular pyramid. I am expecting that you are going to show “graphically” that a triangular prism consists of 3 triangular pyramids. After some graphical tinkering, I come out with the following: Consider 2 triangles, ABC at the bottom (A,B are front angles, C is back angle) and DEF at the top with corresponding angles. Connect vertical lines AD, BE, CF. Also connect diagonal lines AE, AF, BF. Triangle ABC has depth “d” which is perpendicular to AB. Furthermore, AB=w, AD=h. The first pyramid has base ACB and vertical edge CF. The second pyramid has base FDE and vertical edge DA. The third pyramid requires some visualization as it has a “slanted” apex F. It has base AEB and slanted edge EF, but with vertical height equal to “d”. All the 3 pyramids have equal volume as along as their “wdh” products are the same. For the third pyramid, d becomes the height, and h becomes the depth.
Nice to see your input! Always interested to hear what others think! I put out the video because it was fun to go through just geometry for the proof and do the general case. Happy learning!
Love your message! I posted the cone video as well :) … you can find it here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-9YypvVUlUG4.html … welcome to 1mjourney.com
