Miter for a PyramidDate: 07/31/97 at 10:44:04 From: Clive Champion Subject: Miter of a pyramid I am an artist and have a project where I have to construct a 4-sided pyramid out of glass for a garden fountain. Each of the four triangles has a base of 30.5 inches and a height of 46 inches. I need to know what degree of miter to put on the edges of the uprights of the triangles so that it will all fit together nicely. Thanks for any help you can give me. I have come up with 54.125 degrees on each of the upright edges. I am sure there must be a formula but I cannot find it. Clive@artistcraft.com http://www.artistcraft.com Date: 07/31/97 at 13:24:50 From: Doctor Ken Subject: Re: Miter of a pyramid Hi Clive - I'll give your measurements some names: let B be the length of the base of each triangle, and let H be its height. Here's a diagram: First, find the length L of the legs (the sides that aren't the base) of each triangle. You can either measure it, or use the Pythagorean theorem: (B/2)^2 + H^2 = L^2, so L = Sqrt{(B/2)^2 + H^2} We'll use this length L in our calculations. Draw a diagram of your assembled pyramid, and make sure you can see two adjacent faces of it. Locate the midpoints of two of the bases of the triangles: call them m and M. Now draw the line segment connecting them - it's called Mm. Let's name the point connecting the adjacent bases C, and the top of the pyramid T. Now drop a perpendicular from m to TC - it meets TC at J. Note that MJ is also perpendicular to TC (in a different plane). We want to find the measure of angle mJM. To do that, first let's find out everything there is to know about triangle mJM. If we find out the length of its sides, we'll be able to find out everything else. Actually, I'll let you find out the length of its sides - to find out mM, notice that it cuts across the square base at a 45 degree angle. To find mJ (and MJ, since they're the same length), notice that triangle mJC is similar to triangle TmC, so you can set up some ratios. Now that you have these lengths, you can use the law of sines to find the angle mJM. The law of sines says that in a triangle ABC (with side a opposite angle A, b opposite B, and c opposite C), Sin A Sin B Sin C ----- = ----- = ----- a b c Using that, and the fact that the angles add up to 180, you should be able to find out angle mJM, the angle at which two faces of your pyramid meet. If you can't follow this, or get stuck somewhere, write back. Good luck! -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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