Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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.
|Start with a square sheet folded diagonally in half.|
|Fold the two bottom corners together|
|Unfold the triangle.|
|Mark two points as shown. One is 1/3 of the way from the top to the bottom of one edge, the other is 1/3 of the way from the bottom to the top of the other edge.|
|Fold the paper so the two marks coincide and make a crease.|
|Fold the exposed corner up onto|
|Bisect the wedge by folding.|
Finding the edge lengths is a somewhat tedious exercise in similar triangles. In the diagram above, corresponding edges and angles are colored. If we define the height of the triangle as unity, we can derive the four equations shown at lower right in the figure.
We have four equations in four unknowns. The first two equations are simple sums of edge lengths. The last two are based on the proportionality that AO = 1 and BD = √2/3. Let's derive the edges AE and OE for the largest (bottom) triangle. We do this by substituting for and eliminating EB and ED:
Now we have two equations in two unknowns. We can eliminate EA by multiplying the bottom equation by √2/3 and subtracting from the top equation:
Substituting into formula (2) we get
What we really need is EG, but that's simple:
Thus we have EG = 3/7, OG = 4/7 and OE = 5/7. Triangle OEG is a 3-4-5 right triangle. Angle O (blue) is arctan 0.75 = 36.86 degrees, not a bad approximation to 180/5.
Created 22 March 2006, Last Update 14 December 2009