Rhombus
GeometryA 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. It looks like a tilted square. A square is a special rhombus where all angles are also $90^\circ$.
Example
A diamond shape on a playing card is a rhombus. 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.
Key Insight
Rhombus means "spinning top" in Greek. The diagonals of a rhombus always cross at right angles - that is a special property. You can find the area by multiplying the two diagonal lengths and dividing by $2$.
Definition
A rhombus is a parallelogram with all four sides congruent. Properties: opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other at right angles, and each diagonal bisects the vertex angles. Area $= (d_1 d_2)/2$ or base $\times$ height.
Example
Rhombus with side $10$ and one angle $60^\circ$: opposite angles are also $60^\circ$, other pair are $120^\circ$. Diagonals: shorter $= 10$ (= side, since the $60^\circ$ triangle is equilateral), longer $= 10\sqrt{3}$. Area $= (10 \times 10\sqrt{3})/2 = 50\sqrt{3}$ approximately $86.6$.
Key Insight
The perpendicular diagonals of a rhombus are what make the diagonal area formula work: the four right triangles formed by the diagonals are congruent, and the two triangles on each side of a diagonal fit together to make a rectangle of area $d_1 d_2/2 \times 2 = \ldots$ simplified to the known formula.
Definition
A rhombus is a parallelogram with $|u| = |v|$ (equal adjacent side vectors). Its diagonals $u+v$ and $u-v$ are perpendicular because $(u+v)\cdot(u-v) = |u|^2 - |v|^2 = 0$. The symmetry group of a non-square rhombus is $\mathbb{Z}_2 \times \mathbb{Z}_2$ (two reflections across the diagonals, one $180^\circ$ rotation). A rhombus is the intersection of the parallelogram and kite classes.
Example
Rhombus with $u = (3, 0)$ and $v = (0, 3)$ (square): diagonals are $(3,3)$ and $(3,-3)$, perpendicular (dot product $= 9-9 = 0$). Area $= (3\sqrt{2})(3\sqrt{2})/2 = 18/2 = 9 = 3^2$. Confirmed.
Key Insight
The rhombus sits at the intersection of two quadrilateral classes: parallelograms (parallel opposite sides) and kites (two pairs of adjacent equal sides). A rhombus is both a special parallelogram and a special kite, making it a central node in the quadrilateral hierarchy. A square is a rhombus that is also a rectangle.