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Mathematics Forum
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| SleepingDragon Quote | Reply | | Irregular Hexagonal Antiprisms posted on: 5/3/2008 1:53:23 PM If a hexagonal antiprism was made with one regular hexagon and another irregular hexagon how could the resulting distortion of the angles and the lengths of the sides of formerly equilateral triangles connecting the two hexagons together be calculated? On the irregular hexagon four of the sides are the same length as all the sides of the regular hexagon. The other two sides are longer and equal to each other. The shorter sides are 3.53' in length and the longer sides are 3.89' in length. What equation or series of equations, if they exist, could tell me every length and every angle of an irregular hexagon if I knew the lengths of the sides of both hexagons? Does such an equation exist? What follows is my attempt to use trigonometry to find the lengths and angles of the triangles. I have drawn the ring of triangles out in two dimensions, which lengths and angles will be equivalent, that there will be two equilateral triangles present and their positions. There are only four different lengths present and I know two of them so it shouldn't be difficult to find the others and with all the lengths find all the angles. There are four acute isosceles triangles and two lengths of all of them are 3.53'. If I made a right triangle by cutting it in half I get a triangle with 3.53' as the hypotenuse and the other two lengths are the height (unknown) and .5x (unknown). If I only know one angle in this triangle, that it is right and don't know two of the sides but know the hypotenuse is there any possible way to calculate these unknown lengths and angles? Could I solve such a thing purely geometrically by assuming the sides of this strip of triangles are perfectly straight? If this was true there are places where a 60 degree angle added to two equal angles must add up to 180 degrees meaning these two angles are also 60 degrees. That would mean one of the missing angles I would need is 30 degrees, but is it really 30 degrees? Can I assume that the sides are straight if this is an irregular hexagonal antiprism? I know they would be straight in a regular antiprism. Would AutoCAD or a similar computer program be able to automatically calculate these lengths and angles? |
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