5-Pointed Star: Approximate Method 1

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green BayXbr>

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.

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i>Created 22 March 2006, Last Update20 January 2020/i>/p>
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.
 
 
 

Why It Works

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:

  1. EA + OE√2/3 = 2√2/3
  2. EA√2/3 + OE = 1

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:

  1. EA + OE√2/3 = 2√2/3
  2. 2EA/9 + OE√2/3 = √2/3
  3. 7EA/9 = √2/3 or EA = 3√2/7

Substituting into formula (2) we get

  1. (3√2/7)(√2/3) + OE = 1
  2. OE = 1 - 2/7 = 5/7

What we really need is EG, but that's simple:

  1. EG = AE√2/2 = (3√2/7)(√2/2) = 3/7
  2. Since OA = 1, and AG = EG, OG = 1-3/7 = 4/7

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.

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Created 22 March 2006, Last Update20 January 2020