## Properties of a Rhombus

This tool calculates the basic geometric properties of a rhombus (also called diamond shape). Enter below the shape dimensions. The calculated results will have the same units as your input. Please use consistent units for any input.

 Known data: Sides and angle Diagonals Geometric properties: Area = Perimeter = Lengths: Side α = Diagonal p = Diagonal q = Height h = Angles : deg rad φ1 = φ2 = Inscribed circle: Radius r =

## Definitions

### Geometry

Rhombus (also called diamond shape) is a quadrilateral shape with all four sides equal. Pairs of opposite sides are parallel and pairs of opposite angles are equal. Therefore rhombus is also a parallelogram and features all parallelogram properties. It differs from square in its interior angles which are not all equal and 90°.

The area of a rhombus is given by the formulas:

$\begin{split} A & = ah & \quad \textrm{or...}\\ A & = a^2\sin{\varphi_1} & \quad \textrm{or...}\\ A & = a^2\sin{\varphi_2} \end{split}$

where a the length of the sides and h the height, perpendicular to a side from an opposite vertex. Height h can be found, using any of the right triangles, with hypotenuse α shown in figure below:

$\begin{split} h & = a \sin{\varphi_1} & \quad\textrm{or...}\\ h & = a \sin{\left(\pi -\varphi_2\right)} = a \sin{\varphi_2} & \end{split}$

Interior angle φ2 is supplementary with φ1 . Therefore:

$\varphi_2 =180^{\circ} -\varphi_1$

Diagonals p and q of rhombus are mutually bisecting each other, and they also bisect the interior angles φ1 and φ2 . Diagonals can be expressed in terms of side lengths and interior angles as:

$\begin{split} & p = 2a \cos{\frac{\varphi_1}{2}} = 2a \sin{\frac{\varphi_2}{2}}\\ & q = 2a \sin{\frac{\varphi_1}{2}} = 2a \cos{\frac{\varphi_2}{2}} \end{split}$

The perimeter of a parallelogram is simply the sum of the lengths of all sides:

$P = 4a$

The radius of the inscribed circle, can be determined, using the right triangle, with hypotenuse $$\frac{p}{2}$$ (see figure below):

$r = \frac{p}{2} \sin{\frac{\varphi_1}{2}}$