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Infinitely many all different:

As it is only the area that matters, the perimeter can be any shape:

  • An equilateral triangle (with sides approx 3.398 units)
  • A square (with sides approx 2.236 units)
  • A regular pentagon (with sides approx 1.705 units)
  • A regular hexagon (with sides approx 1.387 units)
  • A regular heptagon
  • A regular octagon
  • A regular nonagon
  • A regular decagon
  • A regular hendecagon (11 sided polygon)
  • A regular dodecagon (12 sided polygon)
  • A regular triskaidecagon (13 sided polygon)
  • A regular 14 sided polygon
  • A regular 15 sided polygon
  • A regular 16 sided polygon
  • ...
  • A circle (with radius approx 1.262 units)

And there are also the non-regular shapes, eg an L shaped hexagon, a kite, a parallelogram which can have an area of 5 square units.

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Q: How many shapes can you make with an area of 5 square units?
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