Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay
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The smallest angle of a 3-4-5 right triangle is 36.86 degrees, less than a one-degree error. There are a couple of approximate methods for folding five-pointed stars based on this fact; the astonishing thing is that the constructions never explicitly call for construction of a 3-4-5 right triangle. In fact the numbers 3, 4, or 5 never appear at all in the dimensions of this construction.
/p> </p>Start with a square sheet folded diagonally in half. |
Fold the two bottom corners together |
Unfold the paper. |
Fold the paper lengthwise in half |
Unfold the paper. |
Fold the right edge of the paper until the midpoint of the edge lines up with the top edge as shown. Approximate constructions just don't get any simple than this. |
Fold the exposed corner up onto |
Bisect the wedge by folding. |
Extend the folded center line and construct a line parallel to the tilted end of the paper. The two colored triangles are obviously similar, and we have two equations in two unknowns.
So A = 3/8, the second side = 1/2 = 4/8 and C = 5/8. The triangle is a 3-4-5 right triangle. Furthermore, the blue triangle has edges 1-A = 5/8 and 1-C = 3/8, and must be congruent to the yellow triangle.
The magenta angles are arctan 0.75 = 36.86 degrees, not a bad approximation to 180/5.
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Created 22 March 2006, Last Update 14 December 2009