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One interior angle in a regular triacontagon is 168 degrees, meaning that one exterior angle would be 12°. The triacontagon is the largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons: 168° is the sum of the interior angles of the equilateral triangle (60°) and the regular pentagon (108°).
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 ...
For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem [4]: p. 72 (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n = 3 case. [5] [6]
Sturm's theorem (theory of equations) Sturm–Picone comparison theorem (differential equations) Subspace theorem (Diophantine approximation) Supersymmetry nonrenormalization theorems ; Supporting hyperplane theorem (convex geometry) Swan's theorem (module theory) Sylow theorems (group theory) Sylvester's determinant theorem (determinants)
Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple ...
For each edge, the interior points are all on the same side of the line that the edge defines. The angle at each vertex contains all other vertices in its edges and interior. The polygon is the convex hull of its edges. Additional properties of convex polygons include: The intersection of two convex polygons is a convex polygon.
(The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel.) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry. [7]
The most recent, 9th edition of the SI Brochure does not mention the gradian at all. [22] The previous edition mentioned it only in the following footnote: [26] The gon (or grad, where grad is an alternative name for the gon) is an alternative unit of plane angle to the degree, defined as (π/200) rad. Thus there are 100 gon in a right angle.