Polygons Keystone Geometry

Polygons Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. These figures are not polygons These figures are polygons

Classifications of a Polygon Convex: No line containing a side of the polygon contains a point in its interior Concave: A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

Classifications of a Polygon Regular: A convex polygon in which all interior angles have the same measure and all sides are the same length Irregular: Two sides (or two interior angles) are not congruent.

Polygon Names Classified by the number of sides Triangle 4 sides Quadrilateral 5 sides Pentagon 6 sides Hexagon 7 sides Heptagon 8 sides Octagon 9 sides Nonagon 10 sides Decagon Dodecagon 12 sides n sides n-gon

Convex Polygon Formulas….. Diagonals of a Polygon: A segment connecting nonconsecutive vertices of a polygon For a convex polygon with n sides: The more sides a polygon has, the more diagonals it will have. A square has 2 diagonals. A pentagon has 5 diagonals. A hexagon has 9 diagonals………

Examples Diagonals in a square, n = 4 Diagonals in an octagon, n = 8 ** What about a triangle? A triangle does not have ANY diagonals. Diagonals in an octagon, n = 8

Polygon Angle Formulas Finding the INTERIOR angles in a convex polygon with n sides: The sum of the interior angles is The measure of one interior angle is Finding the EXTERIOR angles in a convex polygon with n sides: The sum of the exterior angles is The measure of one exterior angle is

Solving for the Angles Polygon Triangle Quadrilateral Pentagon Hexagon Interior Angle Sum 180 360 540 720 Each Interior Angle 60 90 108 120 Exterior Angle Sum Each Exterior Angle 72

Solving for the Angles Polygon Heptagon Octagon Nonagon Decagon Interior Angle Sum Each Interior Angle Exterior Angle Sum Each Exterior Angle

Examples….. Sum of the measures of the interior angles of a 11-gon is The measure of an exterior angle of a regular octagon is The number of sides of regular polygon with exterior angle 72 ° is The measure of an interior angle of a regular polygon with 30 sides