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  2. Internal and external angles - Wikipedia

    en.wikipedia.org/wiki/Internal_and_external_angles

    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 ...

  3. Exterior angle theorem - Wikipedia

    en.wikipedia.org/wiki/Exterior_angle_theorem

    In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, [1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel ...

  4. Regular polygon - Wikipedia

    en.wikipedia.org/wiki/Regular_polygon

    The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR 2 where R is the circumradius. [4]: p. 73 The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR 2 − ⁠ 1 / 4 ⁠ ns 2, where s is the side length and R is the ...

  5. Polygon - Wikipedia

    en.wikipedia.org/wiki/Polygon

    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 ...

  6. Simple polygon - Wikipedia

    en.wikipedia.org/wiki/Simple_polygon

    The internal angle of a simple polygon, at one of its vertices, is the angle spanned by the interior of the polygon at that vertex. A vertex is convex if its internal angle is less than π {\displaystyle \pi } (a straight angle, 180°) and concave if the internal angle is greater than π {\displaystyle \pi } .

  7. Triangle - Wikipedia

    en.wikipedia.org/wiki/Triangle

    The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. [34] The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter ...