Parallelogon Calculator

Name: Parallelogon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a convex, hexagonal parallelogon or elongated parallelogram. A parallelogon is a polygon with parallel opposite sides of equal length each, which can be used to seamlessly tile a plane without rotation, so that the edges fit together. There are only four- and six-sided parallelogons. A four-sided parallelogon is called parallelogram, therefore this calculator refers to the hexagonal one.
Enter the three side lengths and two of the angles. The angles must be smaller than 180°. Choose the number of decimal places and click Calculate. Please enter angles in degrees, here you can convert angle units.

Formulas:

α + β + γ = 360 °

d_{a} = \sqrt{b^{2} + c^{2} - 2 b c \cdot \cos (α)}

d_{b} = \sqrt{a^{2} + c^{2} - 2 a c \cdot \cos (β)}

d_{c} = \sqrt{a^{2} + b^{2} - 2 a b \cdot \cos (γ)}

d_{ab} = \sqrt{a^{2} + {d_{a}}^{2} - 2 a d_{a} \cdot \cos [\arccos (\frac{a^{2} + {d_{c}}^{2} - b^{2}}{2 a d_{c}}) + \arccos (\frac{{d_{a}}^{2} + {d_{c}}^{2} - {d_{b}}^{2}}{2 d_{a} d_{c}})]}

d_{ac} = \sqrt{a^{2} + {d_{a}}^{2} - 2 a d_{a} \cdot \cos [\arccos (\frac{a^{2} + {d_{b}}^{2} - c^{2}}{2 a d_{b}}) + \arccos (\frac{{d_{a}}^{2} + {d_{b}}^{2} - {d_{c}}^{2}}{2 d_{a} d_{b}})]}

d_{bc} = \sqrt{b^{2} + {d_{b}}^{2} - 2 b d_{b} \cdot \cos [\arccos (\frac{b^{2} + {d_{a}}^{2} - c^{2}}{2 b d_{a}}) + \arccos (\frac{{d_{b}}^{2} + {d_{a}}^{2} - {d_{c}}^{2}}{2 d_{b} d_{a}})]}

h_{a} = d_{a} \cdot \sin [\arccos (\frac{a^{2} + {d_{a}}^{2} - {d_{ac}}^{2}}{2 a d_{a}})]

h_{b} = d_{b} \cdot \sin [\arccos (\frac{b^{2} + {d_{b}}^{2} - {d_{ab}}^{2}}{2 b d_{b}})]

h_{c} = d_{a} \cdot \sin [\arccos (\frac{c^{2} + {d_{c}}^{2} - {d_{bc}}^{2}}{2 c d_{c}})]

p = 2 \cdot (a + b + c)

v = a + b + d_{c}

w = a + d_{b} + c

x = d_{a} + b + c

y = d_{a} + d_{b} + d_{c}

A = \sqrt{\frac{v}{2} \cdot (\frac{v}{2} - a) \cdot (\frac{v}{2} - b) \cdot (\frac{v}{2} - d_{c})} + \sqrt{\frac{w}{2} \cdot (\frac{w}{2} - a) \cdot (\frac{w}{2} - d_{b}) \cdot (\frac{w}{2} - c)} + \sqrt{\frac{x}{2} \cdot (\frac{x}{2} - d_{a}) \cdot (\frac{x}{2} - b) \cdot (\frac{x}{2} - c)} + \sqrt{\frac{y}{2} \cdot (\frac{y}{2} - d_{a}) \cdot (\frac{y}{2} - d_{b}) \cdot (\frac{y}{2} - d_{c})}

Lengths, diagonals, heights and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The individual values of the parallelogon are calculated by breaking it down into triangles along its diagonals and calculating with these. A parallelogon is point-symmetrical to its center and rotationally symmetrical at an angle of 180 degrees and multiples thereof.
A polyhedron with parallelogons as side faces is called a parallelohedron. Parallelohedrons come in five different versions: cube, hexagonal prism, rhombic dodecahedron, elongated rhombus dodecahedron and truncated octahedron. Cube and rhombic dodecahedrons have parallelograms as side faces, while the other three parallelohedrons contain both parallelograms and hexagonal parallelogons.

Hexagonal parallelogons are created in practice when a regular hexagonal honeycomb structure is tilted or stretched. They are used specifically when a hexagonal structure with a certain asymmetrical property is required. Hexagonal parallelogons are found in some crystals, such as quartz and beryl.

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