Archimedes' Lab Experimental Geometry


Squaring Triangle

The puzzle we'll make together starts as an equilateral triangle, but it unfolds and folds to make a square! Magic? No, simply mathematical! To make our triangular puzzle we don't need any compas or strange tools... We just need a paper sheet, a pencil, a L-shaped square and a pair of scissors.
How to make an equilateral triangle with a paper sheet
Take a standard paper sheet and fold it lengthwise in two equal parts as shown in fig. 2 below. Now, unfold your paper sheet (notice the foldmark in its middle). Then, fold back a corner of the paper sheet on the foldmark and draw a line along the corner side (see fig. 4). Repeat this operation with the opposite corner of the paper sheet. Cut your sheet along the two lines you've drawn (fig. 7). Compliments! You have now a perfect equilateral triangle.
How to transform your equilateral triangle into a square
Fold back the 3 corners of the triangle to make a rectangle as per fig. 9. Then, unfold your triangle. Important: the foldmarks define a rectangle ABCD. Trace the diagonal BD of the rectangle ABCD (fig. 11). Now, with the help of a L-shaped square trace 2 perpendiculars to the diagonal BD, one passing through point A, the other one through point C, as shown in fig. 12. Your triangle is now divided in 4 pieces. Cut them out and reassemble them as shown in fig. 15.

diagram 1
diagram 2
diagram 3
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