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Properties of a Parallelogram
Properties of a Trapezoid
Properties of a Right-Triangle
Properties of a Rectangle
Properties of a Circular area
All Geometric Shapes tools

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:
α =
φ1 =
p =
q =
Geometric properties:
Area =
Perimeter =
Lengths:
Side α =
Diagonal p =
Diagonal q =
Height h =
Angles :
φ1 =
φ2 =
Inscribed circle:
Radius r =
shape details
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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:

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

shape geometry

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):

\[ r = \frac{p}{2} \sin{\frac{\varphi_1}{2}} \]

shape geometry
See also
Properties of a Parallelogram
Properties of a Trapezoid
Properties of a Right-Triangle
Properties of a Rectangle
Properties of a Circular area
All Geometric Shapes tools