Can two figures have the same area, perimeter, and same number of segments have different shape?

In the spirit of the no-words answer to the linked question:

two shapes made up of squares which have the same perimeter and area but which are different shapes


Inspired by Hagen von Eitzen's answer, a particular pair of tetrominoes furnishes another counterexample (minimal among polyominoes):

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No, it is still not enough. Even for quadrilaterals it is not enough. Take a kite with sides $1,1,3,3$ and the angle between the two $1$s a right angle. The perimeter is $8$ and the area is $\frac 12(1+\sqrt{17})$. Now take a rhombus with sides of $8$. It also has a perimeter of $8$ and you can choose the angle to make the areas match. It is even worse for more sides. The area and perimeter are just two constraints, while there are lots of degrees of freedom.

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Geometry

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