Arrow-Hexagon Calculator

Name: Arrow-Hexagon - Geometry Calculator
Author: Jürgen Kummer

Calculations at an arrow-hexagon. An arrow-hexagon here is an isosceles triangle, of which a smaller, similar isosceles triangle is removed at the center of its base. The result is a non-convex hexagon with three pairs of sides. Please enter the angle α and the long side a, as well as one value of b, c or d. Choose the number of decimal places, then click Calculate.
Enter angle in degrees, here you can convert angle units.

Formulas:

c = \sqrt{\frac{d^{2}}{2 \cdot [1 - \cos (α)]}}

e = \sqrt{\frac{2 a^{2}}{1 - \cos (α)}}

e = 2 b + d

h = \sqrt{\frac{4 a^{2} - e^{2}}{4}}

i = \sqrt{\frac{4 c^{2} - d^{2}}{4}}

j = h - i

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

A = \frac{h \cdot e - i \cdot d}{2}

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

The removed similar triangle possesses the same proportions, though a different size, as the original isosceles triangle. This is what geometric similarity simplified means. The area of such an arrow hexagon is equal to the area of the large triangle minus that of the small triangle. Its perimeter is equal to the perimeter of the large triangle, minus the base of the small triangle, plus the lengths of the two legs of the small triangle.
This arrow hexagon exhibits the same symmetry as its underlying triangles. It is axially symmetric with respect to the angle bisector of the apex and the indentation. The arrow hexagon represents the cross-section of a hollow cone. The hollow cone, in turn, is the solid of revolution generated by rotating the arrow hexagon around its axis of symmetry.

There are several geometric shapes that can be referred to as arrows. One of these is the arrowhead quadrilateral, in which the two rear ends also terminate in a single vertex. A simple isosceles triangle is also occasionally termed an arrow or arrowhead, although the term typically refers to non-convex shapes, that is, those featuring an indentation opposite their tip. The tip of such arrows is usually formed by an acute angle. However, this is not a geometric necessity, as it may also be an obtuse angle. Conversely, the two adjacent, symmetrical vertices always form acute angles.

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