Volume of a Cone: Two methods, one answer
[S1 - J1 Maths : Volume of a Pyramid (and for the brave, power sums & integration)]
On our last post, we discussed the method which the volume of a sphere is derived. But, for it to work, we need the volume of a cone to work. We have to decisively prove that the volume of a cone is 1/3 x base area x height.
Thankfully, if you remember the video from the last post, Cavalieri's Principle states that as long as the base area of the shapes are equal at every point along the height, then the shapes have equal volume. So, the road to finding the volume of a cone is actually related to finding the volume of a pyramid; A pyramid with the same base area as the circle will ultimately yield the answer we want.
(recap photo for those who suffer from too-much-homework-therefore-I-cannot-remember-please-do-not-confiscate-my-phone-mum)
I'll be sharing two different videos that explain the proof process carefully (my condolences if you watched the Don't Memorise video which did not bother to explain how the formula came to be, thereby forcing you to memorise and ironically contradicting its channel name). The first video is simple and intuitive to understand:
That was nice and clean, wasn't it? Now, we turn our attention to "the other video" mentioned at the end. While a little tougher than before, it is also good to learn and study the process behind it.
Why? Because learning the processes taken step-by-step is critical to tackling Math question you have never seen before. Here:
Each of the videos show an out-of-the-box method of finding the formulae. And you might be wondering, by this point - Why should I learn how it works? Isn't it enough to know that it works?
I would like to answer, "If you understand what the exam setter is thinking about, you no longer have to worry about which topics to revise." Simply by understanding it elevates you to a field of study much tougher, and much more rewarding than simply being told what to do like a machine.
You have a right to know why something is true. You have a right to ask why something should be true, or false. Why are there people who are neither male nor female? Why is there so much violence on the streets of Brazil and India? Why did the Americans choose a president that can't protect them from Covid-19 when it counts?
While not all teachers can produce great answers for those questions, all great answers are worth being taught. The very idea of thought itself is the most important part of you. You are only human when you think.
"Cogito, ergo sum." - Rene Descartes, Discourse on the Method