Square
GeometryA 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 (all $90^\circ$). It is both a rectangle (four right angles) and a rhombus (four equal sides).
Example
A chessboard square, a sticky note, and most floor tiles are squares. If each side is $5$ cm: area $= 5 \times 5 = 25$ square cm, perimeter $= 4 \times 5 = 20$ cm.
Key Insight
The square is the most symmetrical rectangle. It has four lines of symmetry (two through midpoints of sides, two through corners) and looks the same when rotated by $90^\circ$, $180^\circ$, or $270^\circ$.
Definition
A square is a regular quadrilateral: all four sides congruent and all four angles equal to $90^\circ$. It is simultaneously a rectangle, a rhombus, and a parallelogram. For side length $s$: area $= s^2$, perimeter $= 4s$, diagonal $= s\sqrt{2}$. The diagonals of a square are congruent, bisect each other, and are perpendicular.
Example
Square with side $7$: area $= 49$, perimeter $= 28$, diagonal $= 7\sqrt{2}$ approximately $9.9$. The diagonal divides the square into two isosceles right triangles ($45$-$45$-$90$). The diagonals intersect at $90^\circ$, each bisecting the other.
Key Insight
The formula area $= s^2$ is where the word "square" for exponent comes from: $s^2$ literally means "the area of a square with side $s$." The diagonal $= s\sqrt{2}$ comes from the $45$-$45$-$90$ triangle with legs $s$.
Definition
A square is a regular $4$-gon with symmetry group $D_4$ (dihedral group of order $8$: four rotations and four reflections). It achieves the maximum area among all quadrilaterals with a given perimeter (isoperimetric inequality for quadrilaterals). 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
The four rotational symmetries of a square: $0$, $90$, $180$, $270$ degrees (multiples of $\pi/2$). The four reflection symmetries: through horizontal midline, vertical midline, and two diagonals. These eight symmetries form the dihedral group $D_4$.
Key Insight
The square's connection to the fourth roots of unity (vertices at $\{1, i, -1, -i\}$) is the geometric manifestation of the algebraic identity $z^4 = 1$. This connects the square to complex number theory, and more broadly, regular $n$-gons correspond to the $n$-th roots of unity - a bridge between geometry and abstract algebra.