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$\begingroup$

I've got that the $s=a\sqrt{\frac{2-\sqrt{3}}{3}}$

And $d=\sqrt{2}(1+\sqrt{3})s$

Where $a$ is the side of the square, $s$ is the side of the dodecagon, and $d$ is the diameter of the dodecagon. But i can't go further. I've tried to do something with GeoGebra, but i didn't get anything usefull. I don't need the best solution with the lowest ammount of pieces, just a possibility.

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    Try Google with key words: **dissection + square + dodecagon** – 2017-02-18
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    Did you find anything when you tried it, 145014? – 2017-02-20
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    Are you still here, 145014? – 2017-02-22

2 Answers 2

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You can do it in six pieces:

enter image description here

Take an equilateral triangle as shown in blue, then continue for one edge length on the bisector as shown. Draw circles of radius one edge length as shown and cut out the top piece (a bit like a T). With the triangle these two pieces make a four-point star. Finally cut the side pieces in half along the long axis of symmetry. Those last (green) cuts will be the sides of the square, leaving the star-shaped hole to be filled in the centre.

To make the same process in reverse, draw the diagonals and mid-lines (mid-side perpendiculars) on a square. Then draw an equilateral triangle in one corner, as shown, and continue the side from the corner to the mid-line. Bisect this extended line to intersect the diagonal. This gives the side length of the dodecagon. Use this length to draw the four-point star from mid-lines to digonals and remove a triangle from this star to gives the shapes as per the dodecagon dissection above.

enter image description here

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    thanks, but the starting-point was a square, and the finish was the dodecagon. – 2017-02-18
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    I could make this: http://prntscr.com/eafdax , but i could not really make a square of it :/ What order should i put them together? I've colored the parts: http://prntscr.com/eafhcu – 2017-02-18
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    Arrange the 4 edges from the final 2 long straight cuts as the edges of the squares. – 2017-02-18
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    @user145014 added the square to dodecagon dissection. – 2017-02-20
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    Is this the minimum number of pieces? – 2017-02-20
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    @GerryMyerson I believe so... you have a lot of $150°$ angles to make... – 2017-02-20
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Lindgren, Recreational Problems in Geometric Dissections, revised and enlarged by Greg Frederickson, gives several six-piece dodecagon-to-square dissections, Figures 9.7 through 9.10 on pages 41-42. His Figure 9.9 is the one in Joffan's answer, which Lindgren/Frederickson says "is probably the neatest," but he also recommends the one in his Figure 9.8 in which "the cuts in the dodecagon are so easy – just four chords." I don't have the skills to draw pictures here, but the description is quite simple. Number the vertices of the dodecagon from 1 to 12 in cyclic order, then draw the chords joining both 1 and 12 to both 5 and 8. This cuts the dodecagon into six pieces that can be rearranged to form the square.