POLYGONS Grade 8 POLYGONS What is a polygon? A plane figure bounded by straight lines is called a Polygon. Regular polygons have all sides and angles equal and irregular polygons have some sides and angles different. Convex polygon A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set. In a convex polygon, all interior angles are less than or equal to 180°, while in a strictly convex polygon all interior angles are strictly less than 180°. An example of a convex polygon: a regular pentagon Properties of convex polygons The following properties of a simple polygon are all equivalent to convexity: • Every internal angle is less than or equal to 180 degrees. • Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. • The polygon is entirely contained in a closed half-plane defined by each of its edges. • 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. __________________________________________________________________________________ 1 www.eduinfinite.com POLYGONS Grade 8 Concave polygon A simple polygon that is not convex is called concave, A simple concave polygon will always have an interior angle with a measure that is greater than 180 degrees. It is always possible to partition a concave polygon into a set of convex polygons. An example of a concave polygon. Simple or Complex A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex. Simple Polygon Complex Polygon (this one's a Pentagon) (also a Pentagon) Types of Polygon A polygon with all sides and interior angles the Regular same. Regular polygons are always convex. Each side may a different length, each angle may Irregular be a different measure. The opposite of a regular polygon. __________________________________________________________________________________ 2 www.eduinfinite.com POLYGONS Grade 8 All interior angles less than 180°,and all vertices 'point outwards' away from the interior. The Convex opposite of concave. Regular polygons are always convex. One or more interior angles greater than 180°. Some vertices push 'inwards' towards the interior Concave of the polygon. The opposite of convex. A polygon where one or more sides crosses back over another side, creating multiple smaller Self- polygons. Most of the properties and theorems intersecting concerning polygons do not apply to this shape. It or Crossed is best considered as several separate polygons. A polygon that in not self-intersecting in this way is called a simple polygon. Properties of all Polygons (regular and irregular) Interior The interior angles of a polygon are those angles angles at each vertex on the inside of the polygon. Exterior The angle on the outside of a polygon between Angles a side and the extended adjacent side. Diagonals The diagonals of a polygon are lines linking any two non-adjacent vertices. Area For regular polygons there are various ways to calculate the area. For irregular polygons things are a little harder since there no general formulae. Perimeter The distance around a polygon. The sum of its side lengths. __________________________________________________________________________________ 3 www.eduinfinite.com POLYGONS Grade 8 Consider the angle sum of a quadrilateral. Clearly, the diagonal AC divides the quadrilateral into 2 triangles. Angle sum of a quadrilateral = 2 x 1800 =3600 Consider the angle sum of a pentagon. Clearly, the diagonals AC and AD divides the pentagon into 3 triangles. Angle sum of a pentagon= 3 x 1800 =5400 Consider the angle sum of a hexagon. Clearly, the diagonals AC, AD, and AE divides the hexagon into 4 triangles. __________________________________________________________________________________ 4 www.eduinfinite.com POLYGONS Grade 8 Angle sum of a hexagon= 4 x 1800 =7200 In general: Note that a polygon of n sides is called an n-gon. To Know more, register for EDUINFINITE Classes __________________________________________________________________________________ 5 www.eduinfinite.com .