POLYGONS

Grade 8

What is a ?

A plane figure bounded by straight lines is called a Polygon.

Regular polygons have all sides and equal and irregular polygons have some sides and angles different.

Convex polygon

A is a (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 . 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

Properties of convex polygons

The following properties of a simple polygon are all equivalent to convexity:

• Every internal 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 of its edges.

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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.

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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 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.

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POLYGONS

Grade 8

Consider the angle sum of a .

Clearly, the AC divides the quadrilateral into 2 .

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 .

Clearly, the diagonals AC, AD, and AE divides the hexagon into 4 triangles.

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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.

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