Square inscribed in a circle
Try this Drag the orange dot A.
Note how the four
vertices
of the square always lie on the circle.
A square inscribed in a circle is one where all the four
vertices
lie on a common circle.
Another way to say it is that the square is 'inscribed' in the circle. Here, inscribed means to 'draw inside'.
Diagonals
The diagonals of a square inscribed in a circle intersect at the center of the circle. To see this
check the 'diagonals' box in the applet above. As with all squares, the diagonals bisect each other
at right angles.
Circumcircle
Another way to think of this is that every square has a
circumcircle  a circle that passes through every vertex.
In fact every regular polygon has a circumcircle, and so can be inscribed in that circle.
Construction
You can construct a square inscribed in a circle using compasses and a straightedge. For more see
Constructing an inscribed square.
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Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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