Regular Hexadecagon Calculator

Name: Regular Hexadecagon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a regular hexadecagon, a polygon with sixteen edges and just as many vertices.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

d_{8} = \frac{a}{\sin (\frac{π}{16})}

d_{7} = \frac{\sin (\frac{7 π}{16})}{\sin (\frac{π}{16})} \cdot a

d_{6} = \frac{\sin (\frac{3 π}{8})}{\sin (\frac{π}{16})} \cdot a

d_{5} = \frac{\sin (\frac{5 π}{16})}{\sin (\frac{π}{16})} \cdot a

d_{4} = \frac{\sqrt{2} / 2}{\sin (\frac{π}{16})} \cdot a

d_{3} = \frac{\sin (\frac{3 π}{16})}{\sin (\frac{π}{16})} \cdot a

d_{2} = \frac{\sin (\frac{π}{8})}{\sin (\frac{π}{16})} \cdot a

Höhe = d_{7}

p = 16 a

A = 4 a^{2} \cot (\frac{π}{16})

r_{c} = \frac{d_{8}}{2} = \sqrt{\frac{4 + 2 \sqrt{2} + \sqrt{20 + 14 \sqrt{2}}}{2}} a

r_{i} = \frac{d_{7}}{2} = \frac{1 + \sqrt{2} + \sqrt{2 (2 + \sqrt{2})}}{2} a

Angle: 157.5°
104 diagonals

Edge length, diagonals, height, perimeter and radius have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The regular hexadecagon is a convex, regular polygon with sixteen equal sides, each with equal angles between them. These angles have an internal angle of 157.5 degrees, and an external angle of 202.5 degrees. The regular hexadecagon is point-symmetric about the intersection of the diagonals over eight sides, which are the longest of the seven different types of diagonals. It is also axially symmetric about these eight diagonals, as well as about the eight perpendicular bisectors through the opposite sides. It therefore has sixteen axes of symmetry. A hexadecagon is relatively easy to construct with compass and straightedge. This was already known to ancient Greek mathematicians.
Hexadecagons are sometimes used in art and architecture. They can be found in the floor plans of buildings, which are intended to appear almost, but not quite, round. One such building is the Aachen Palatine Chapel from the Carolingian period, which was built around 800 AD and is part of Aachen Cathedral. This building has a sixteen-sided exterior contour, but the interior is eight-sided, i.e., an octagon. Another sixteen-sided building can be found in the Renaissance painter Raphael's 1504 painting The Marriage of the Virgin. A tower of this type, the product of the painter's imagination, occupies a dominant position in the background.

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