Square
A square is a regular quadrilateral with all four sides equal and all four angles equal to 90 degrees.
Formula
\text{Area} = s^2; \text{Perimeter} = 4s; \text{Diagonal} = s\sqrt{2}
Definition
A square has four equal sides and four right-angle corners, making it both a rectangle (four right angles) and a rhombus (four equal sides); it is a regular quadrilateral, all sides congruent and all angles $90^\circ$. For side length $s$: area $=s^2$, perimeter $=4s$, diagonal $=s\sqrt{2}$. Its symmetry group is the dihedral group $D_4$ of order $8$ (four rotations and four reflections), and it achieves the maximum area among all quadrilaterals with a given perimeter; in the complex plane, the fourth roots of unity $\{1, i, -1, -i\}$ are the vertices of a unit square centered at the origin.
Example
A chessboard square, a sticky note, and most floor tiles are squares; for side $5$ cm, area $=25$ and perimeter $=20$ cm. For side $7$: area $=49$, perimeter $=28$, diagonal $=7\sqrt{2}\approx9.9$, formed by two isosceles right ($45$-$45$-$90$) triangles whose diagonals meet at $90^\circ$. The square's four rotational symmetries ($0, 90, 180, 270$ degrees) and four reflections (through the horizontal midline, vertical midline, and two diagonals) together form $D_4$.
Key Insight
The square is the most symmetrical rectangle, looking the same when rotated by $90^\circ$, $180^\circ$, or $270^\circ$, and the formula area $=s^2$ is literally where the word "square" for an exponent comes from: $s^2$ means "the area of a square with side $s$." The square's connection to the fourth roots of unity ($z^4=1$) is the geometric manifestation of that algebraic identity, and more broadly, regular $n$-gons correspond to $n$-th roots of unity, a bridge between geometry and abstract algebra.