The interior angle is enclosed by two adjacent side edges. The variable is in the general formula for regular polygonsfor the number of corner points of the polygon. In this case the variable is number to use.
The central angle or central angle is made up of two neighboring perimeter radii locked in. In the general formula is for the variable the number to use.
The inscribed radius is the height of a partial triangle, perpendicular to the length of the side of the 51-sided. If the same right-angled triangle is used for the calculation as for the side length, the inscribed radius applies
The height of a regular 51-sided result from the sum of the incircle radius and radius .
The area of a triangle is generally calculated . The results of the side length are used to calculate the 51 gon and the incircle radius used what for the height is used.
from this it follows for the area of a partial triangle
summarized it results
and for the area of the entire 51 gon
As described above in the regular 51-square , the 51-square can be represented as a construction with compass and ruler . Since the number of its vertices results from the multiplication of the two Fermat prime numbers and results, the regular 51-sided can be found by an extension of an already known construction of the 17- sided. The two polygons triangle and seventeen-sided (their number of sides corresponds to the Fermat prime numbers or. ) are placed on top of each other symmetrically with regard to their central angles in the common circumference, as is the case with z. B. Johannes Kepler in his work WORLD HARMONIK in the construction of Fünfzehnecks shows  .
In principle, one of the three methods described in Siebzehneck can be selected as the basis for the construction . For reasons of the very little effort required, the method of Duane W. DeTemple ,  from 1991, is used.
Fig. 1: Seventeen-corner after Duane W. DeTemple
In the drawing of the seventeenth-corner by Duane W. DeTemple (Fig. 1) it is easy to see the perpendicular from the center not only cuts the arc of a circle but also the periphery. If this intersection is called marked, it is right next to the corner point This results in the central angle with the angular widthof an equilateral triangle , which is added geometrically clockwise to the central angle of the seventeenth corner.
Thus the route is one side length and a corner point of the 51 corner you are looking for.
The position of the corner point of the 51-sided result from the number of side lengths those in the central angle are included
starting from the corner point not counted clockwise the 17th corner corresponds to the corner which is counted counterclockwise.
The 17th corner point of the 51 corner is therefore based on the central axis , exactly opposite the 34th corner point.
The identifiers in Figure 2, which have been changed compared to the original, correspond to those used today.
Fig. 2: Extension of the construction of the 17-sided according to Duane W. DeTemple to the construction of the 51-sided by adding the corners of the equilateral triangle ( - - ) and removing the missing points
Draw a straight line (analytically an X-axis) and determine a point on the later center point of the polygon (analytically a coordinate origin).
Drawing a circle as a perimeter (analytically a unit circle) one . There are two points of intersection , the corner point of the polygon and the counterpoint .
Establishing the vertical (analytically a Y-axis) on the straight line in . The point of intersection results.
Halving the route in .
Establishing the vertical on the straight line in . The two intersections with are the cornerstones and des 51-Ecks.
Draw the circular arc one with the radius . The intersection with the vertical is.
Now is around the first Carlyle circle through the point drawn; the intersections are and .
The distance is halved. You get.
Draw a second Carlyle circle one by . The points of intersection with x are the points and (the latter is not shown because it is no longer needed).
The distance is halved. You get.
Draw a third Carlyle circle one by . The points of intersection with x are the points and (The latter is also not shown, as it is no longer needed).
Removal of the route on of off off. You get point
Connect the dots and with a route.
Halve the route . You get point.
Draw a fourth Carlyle circle one by . The points of intersection with x are the points and (the latter not labeled as it is no longer needed).
Redraw an arc of a circle with the perimeter radius . The intersections with the perimeter are the two too adjacent points of the 17-sided and thus the points and des 51-Ecks.
By repeatedly removing the route on the perimeter , starting with , you get the missing points of a regular 17-sided. Up to this point the construction corresponds to that of the 17-sided.
By repeatedly removing the route on the perimeter , starting from the points (blue) and (red), you get all the missing corner points of the 51 corner, which can be connected to each other to form the 51 corner.
RWE tower in Essen
The cross-section of the RWE tower in Essen is a regular 51-sided.
↑ Johannes Kepler: WORLD HARMONICS. XLIV. Sentence., Page of the Fifteenth, Page 44. In: Google Books. R. OLDENBURG VERLAG 2006, translated and introduced by MAX CASPAR 1939, p. 401 , accessed on February 21, 2018 .