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No.

The total of the interior angles of a polygon is given by:

total_of_interior_angles = 180° × (number_of_sides -2)

A regular polygon has every interior angle the same, and the size of each angle is given by:

size_of_interior_angle = total_of_interior_angles / number_of_sides

→ size_of_interior_angle = 180° × (number_of_sides -2) / number_of_sides

But as number_of_sides - 2 is (always) less than number_of_sides, (number_of_sides -2) / number_of_sides is less than 1 and so the size_of_interior_angle is less than 180°.

To be a concave polygon, at least one interior angle must be greater than 180°.

But as a regular polygon has all interior angles less than 180°, no regular polygon can be concave.

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Q: Can you ever have a polygon that is regular and concave?
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A square is an example of a quadrilateral that is both a regular concave polygon and a regular convex polygon?

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