a square
/ skwair /
A square is the most familiar 'perfect' four-sided shape — a chessboard cell, a floor tile, a sticky note — with four equal sides and four right-angle corners. It is the quadrilateral nothing can be made more even: turn it a quarter-turn and it lands exactly on itself.
Precisely, a square is a quadrilateral that is both equilateral (all four sides equal) and equiangular (all four angles right angles), which makes it the regular 4-gon. Equivalently — and this is the point of the family hierarchy — a square is a rectangle with all sides equal, and also a rhombus with all angles right; it is exactly the figure that is simultaneously a rectangle and a rhombus. It therefore collects every special property of both: opposite sides parallel and equal, all angles 90 degrees, and diagonals that are equal (the rectangle gift), perpendicular (the rhombus gift), bisect each other, and bisect the corner angles into 45-degree halves.
The square sits at the very top of the quadrilateral hierarchy, the single shape that satisfies every defining condition below it. The classic trap is direction of inclusion: every square is a rectangle and a rhombus and a parallelogram and (under the inclusive definition) a trapezoid, but the reverse fails — most rectangles and rhombi are not squares. So a true statement like 'a square is a rhombus' does not reverse to 'a rhombus is a square'. Keep the one-way arrows of the hierarchy straight.
A square with side 1 has diagonals of length sqrt(2), equal to each other, meeting at the centre at right angles, and bisecting the 90-degree corners into 45-degree angles. It is at once a rectangle (equal diagonals) and a rhombus (perpendicular diagonals).
The square inherits both the rectangle's and the rhombus's diagonal properties.
Inclusion runs one way: every square is a rhombus and a rectangle, but most rhombi and rectangles are not squares. 'A square is a rhombus' is true; its converse is false.