Symmetric Hexagon Calculator

Name: Symmetric Hexagon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a symmetrical hexagon, a hexagon that is axisymmetric to its first diagonal and its height (d and h) and point symmetric to their intersection. This consists of two equal isosceles trapezoids that are joined on one of their two parallel sides. The opposite sides of the symmetrical hexagon are parallel to each other.
Enter the two side lengths and the length of the first diagonal, choose the number of decimal places and click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.

Formulas:

h = \sqrt{4 a^{2} - {(d - b)}^{2}}

e = \sqrt{b^{2} + h^{2}}

α = \arccos (\frac{h}{2 a}) (convex shape)

α = - \arccos (\frac{h}{2 a}) (concave shape)

p = 4 a + 2 b

A = \frac{b + d}{2} \cdot h

Side lengths, diagonals, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The angle α is the angle at the left and right vertices. If the angle is calculated as a negative value, then the symmetrical hexagon is concave, i.e., curved inward, as in the second image.

Another possible name for this shape is an isosceles hexagon. The four equal sides of the convex variant can be thought of as the legs of two equal, mirrored isosceles triangles, with a matching rectangle between them. The elongated regular hexagon is also such a symmetrical or isosceles hexagon, just as the regular hexagon of course.

If you imagine this hexagon as composed of two equal isosceles trapezoids, you can simply swap these two trapezoids to convert the convex shape into the concave shape, and vice versa. Therefore, the formulas for these two variants of the symmetrical hexagon are identical, except for the one for the angle at the junction of the two trapezoids.

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