Pyramid ConstructionDate: 05/24/97 at 02:39:31 From: Don Mohr Subject: Pyramid building I'm not a student - I am a professor of history. My son is an advanced math tenth grader. Neither he nor his teacher could help me. I need to construct a four-sided pyramid for a children's exhibit at the Anchorage Museum. I want to use plywood. I need to figure out the compound angles (angle of side, and mitre) that will join the sides. After a much-needed review of trigonometry (and with my son's help), I was able to figure out the mitre angle needed. The angle of the sides of the pyramid are directly calculable by the Pythagorean theorem. I found the mitre angle (joints between sides) by laying out a right angle triangle on a side one foot from the point of the base. Knowing the side angle, one foot length of side, and ninety degree angle allows the calculation of the other sides. Then looking down at a corner of the base from above the pyramid, one has a triangle that is perpendicular to the joining edge of two sides. The length of all sides of this triangle are known. The angle of the joined sides may be calculated and the mitre revealed. My son has even programmed his TI-85 to calculate the compound mitre of any four-sided pyramid by entering only the base length and height. Now my question is this: It seems to me that there should be a more direct path to calculating the angle of the sides. It appears that the angle of the sides must change in a linear fashion determined by the base length and height. Is there a formula that will do this? Once this angle is known, the mitre calculation can be done in one's head. Thus, one would not have to carry around a TI-85. Thanks for your help. Date: 05/24/97 at 07:57:10 From: Doctor Mitteldorf Subject: Re: Pyramid building Dear Don, A picture would really help here! I gather that the pyramid you are working with has 4 triangular faces and a square base (as opposed to a tetrahedron, which has 4 triangular faces, including the base). A square pyramid like this can be constructed with four copies of any isoceles triangle, as long as the height of the triangle is more than half its base. Once you decide the dimensions of the isoceles triangle, you can use the fact that the apex is above the center of the square base to calculate the mitre angle of the base: the cosine of this angle is half the base of the triangle divided by the height of the isoceles triangle. The only other angle left to compute is the mitre angle where any two adjacent triangular faces meet. There's a formula I carry around in my head that is useful in this and many situations like it, involving angles between lines and between planes. I'll describe it to you, with three special cases: Open a notebook and draw a line from a point on the book's spine diagonally across one page. Draw another line starting at the same point on the spine, going diagonally across the facing page at a different angle. Let's say that the angle between the spine and the first line is a and the angle between the spine and the second line (measured on the facing page) is b. Let x be the angle between the two lines. Now this angle x will depend on how wide open the book is. If the book is open flat, then the angle x becomes a + b. If the book is closed tight, the angle x becomes a - b. The interesting case, of course, is when the book is open to an intermediate angle. If x is a right angle so that the book is open to 90 degrees, then cos(x) = cos(a)*cos(b) You can write the other two cases in terms of cosines as well: cos(x) = cos(a)*cos(b) - sin(a)*sin(b) Book open flat cos(x) = cos(a)*cos(b) + sin(a)*sin(b) Book closed tight These come from the formulas for the cosine of the sum and difference of two angles. Now here comes the general case: For any angle c that the book may be open: cos(x) = cos(a)*cos(b) + sin(a)*sin(b)*cos(c) Notice that this formula subsumes the three special cases we did first. The general formula can be derived easily using vector dot products. It can also be derived straight from trigonometry, if you have a lot of imagination and are good at visualizing 3-dimensional objects. I leave it to you to calculate your pyramid angles using this formula. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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