5-Pointed Star: Approximate Method 2

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof 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. In fact the numbers 3, 4, or 5 never appear at all in the dimensions of this construction.

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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 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.

Why It Works

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.

  1. (1-A)/c = (1-C)/a
  2.  A(1-A) = C(1-C)
  3. A - A2 = C - C2
  4. A - C  = A2 - C2 = 1/4
  5. C = A + 1/4
  6. A - A2 = C - C2 = A + 1/4 - (A + 1/4)2
  7. A - A2 = A + 1/4 - A2 - 2(1/4)A - 1/16
  8. 0 = 1/4 - A/2 -1/16
  9. A/2 = 3/16 or A = 3/8
  10. C = A + 1/4 = 5/8

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 Update20 January 2020