A lozenge, commonly known as a rhombus, is a special type of parallelogram where all four sides are of equal length. Unlike a square, the angles are not necessarily right angles. Lozenge shapes are symmetrical and often appear in tiling, design patterns, and geometry.
The total length around the lozenge:
\[ P = 4a \]
where:
The area can be calculated using either the diagonals or base and height:
\[ A = \frac{m \times n}{2} = a \times h \]
where:
The sum of adjacent angles is always \(180^\circ\):
\[ \alpha + \beta = 180^\circ \]
Relationship between side length and diagonals:
\[ m^2 + n^2 = 4a^2 \]
Derived using the Pythagorean theorem, as diagonals intersect at right angles.
Height can be computed using two different formulas:
\[ h = \frac{mn}{2a} = a \sin(\alpha) \]
where: