Volume of a Regular Octahedron
Date: 09/12/2004 at 10:50:56 From: Jane Subject: (no subject) How do you find the volume of a regular octahedron, if you don't know the height of one of the pyramids? For example, find the volume of a regular octahedron with sides of 1 cm. I know volume = (2/3)AH, but I can't find 'H' and my answer never seems right.
Date: 09/12/2004 at 13:02:05 From: Doctor Ian Subject: Re: (no subject) Hi Jane, If it's a _regular_ octahedron, then you can divide it into two pyramids, where each pyramid will have all its edge lengths the same. Does that make sense? So, you have a regular pyramid with all edges of length L, and you want to find the volume. To use the formula L^2 h V = ----- 3 you need to find the height. To find the height, you'll need to use the Pythagorean theorem. To use the Pythagorean theorem, you need to have a right triangle where you know two of the sides. Here is such a triangle: One vertex, A, is at a base corner of the pyramid. A second vertex, B, is at the apex of the pyramid. The third vertex, C, is at the center of the base. You know that AB is just the length of an edge, which is L. BC is what you're trying to find. So you need to know AC, the distance from a corner of the base to the center of the base. L A-----------+ | . | | . | | C | L | | | | +-----------+ How do you find this? You use the Pythagorean theorem again! As you move on in math, you're going to stop seeing problems where you can just whip out a formula, fill in the values, and get the answer. More and more, you're going to see problems that you'll have to reduce to smaller and smaller problems, get the answers to the smaller problems, and put them back together to get the final answer you're looking for. In this case, the first thing we do is break the problem, What is the volume of an octahedron...? into the smaller problem, The volume of an octahedron is twice the volume of a corresponding pyramid. So what is the volume of one of the pyramids? For this, we have a formula: L^2 h V = ----- 3 but to _use_ the formula, we need to know the height. So we have to find the height. If we knew the distance from a corner to the center, we could find the height. So now we have to find the distance from the corner to the center. And then we can just unwind it all: Using the distance from the corner to the center, we can find the height. Using the height, we can find the volume of the pyramid. Using the volume of the pyramid, we can find the volume of the octahedron. In this case, we had a formula that we could use for the volume of the pyramid. In many cases, there won't be any such formula. You'll have to take a detour and invent one for yourself--inventing the formula will be one of your sub-problems, and that will break into sub-problems, which will break into more sub-problems, and so on. This kind of problem reduction, by the way, is the MAIN skill that you're supposed to be learning in your math classes. All the earlier stuff about learning to multiply and divide, and then using variables, and so on, is just preparation for this kind of problem-solving. In other words, it's kind of like: Okay, now you can stand up, and walk, and even run a little. Now it's time to actually learn baseball, or soccer, or whatever. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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