Rhombus
A rhombus is a parallelogram with all four sides of equal length, with perpendicular diagonals that bisect each other.
Formula
\text{Area} = (d_1 d_2)/2 \text{ (where } d_1, d_2 \text{ are diagonal lengths)}
Definition
A rhombus is a four-sided shape where all four sides are the same length, like a tilted square; a square is the special rhombus where all angles are also $90^\circ$. Formally, it is a parallelogram with $|u|=|v|$ for its adjacent side vectors, whose diagonals $u+v$ and $u-v$ are perpendicular because $(u+v)\cdot(u-v)=|u|^2-|v|^2=0$; area $=(d_1 d_2)/2$, and its symmetry group (for a non-square rhombus) is $\mathbb{Z}_2\times\mathbb{Z}_2$.
Example
A diamond shape on a playing card is a rhombus, and a rhombus can look like a square leaning to one side; if each side is $5$ cm, it is a rhombus regardless of the angles. With side $10$ and one angle $60^\circ$: the opposite angle is also $60^\circ$, the other pair $120^\circ$, the shorter diagonal equals the side ($10$), the longer diagonal is $10\sqrt{3}$, giving area $(10\times10\sqrt{3})/2=50\sqrt{3}\approx86.6$. For $u=(3,0)$, $v=(0,3)$ (a square): diagonals $(3,3)$ and $(3,-3)$ are perpendicular, with area $=(3\sqrt{2})(3\sqrt{2})/2=9=3^2$.
Key Insight
The word "rhombus" means "spinning top" in Greek, and its diagonals always cross at right angles, a property that makes the area formula (multiply the diagonals and divide by $2$) work. The rhombus sits at the intersection of two quadrilateral families: parallelograms (parallel opposite sides) and kites (two pairs of adjacent equal sides), making it a central node in the quadrilateral hierarchy, and a square is simply a rhombus that is also a rectangle.