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The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...
However, it is constructible using neusis, or an angle trisector. The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle O A ¯ = 12 , {\displaystyle {\overline {OA}}=12,} according to Andrew M. Gleason , [ 1 ] based on the angle trisection by means of the Tomahawk (light blue).
Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, [11]: p. xi and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.
For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126°. To find the number of sides this polygon has, the result is 360 / (180 − 126) = 6 2 ⁄ 3, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icositetragon , m =12, and it can be divided into 66: 6 squares and 5 sets of 12 rhombs.
Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector. Close approximations to the regular hendecagon can be constructed. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units ...
Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 ... A hexadecagram is a 16-sided star polygon, ...
If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. Two congruent shapes are similar, with a scale factor of 1. However, some school ...