The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: Math Kangaroo 5-6, Math Kangaroo 7-8, SAT-I, SAT-II, GMAT, AMC-8.

Definition: a closed, non self-intersecting plane figure formed of segments.

Convex and concave:

Any two points chosen inside a convex polygon can be connected by a segment that lies entirely inside the polygon.

In a concave polygon, two points can be found such that the segment that connects them crosses the polygon's boundary and lies partly outside the polygon.

figure

Many of the concepts used in problems are valid only for convex polygons. The problem may state that the polygon is regular just in order to ensure that the student understands it to be convex. However, such a condition is much stronger than convexity.

Various elements of a polygon:

figure

Sum of the interior angles of a convex polygon with N sides

Choose a point P arbitrarily inside the polygon. Divide the polygon in triangles. The sum of the interior angles of each triangle is 180°. There are N triangles.

figure

The total sum over all triangles is:

equation

However, this includes the 360° angle around the arbitrary point P which was used to divide into triangles. We must subtract this quantity:

equation

equation

Sum of the exterior angles of a convex polygon with N sides

The exterior angles of a convex polygon always add up to 360°.

figure

Any exterior angle is the supplement of an interior angle:

equation

where x1 is an exterior angle and a1 is a corresponding interior angle.

For all N angles:

equation

equation

equation

equation

The perimeter of a polygon is the sum of the lengths of all its sides.

The perimeter and the area of an arbitrary polygon can be calculated by dividing it into triangles and by applying the geometry of the triangle.

Regular Polygons are polygons with all sides congruent and all angles congruent.

Any regular polygon is convex.

The interior angle of a regular polygon can be calculated by dividing the sum of the interior angles by the number of angles (since they are all congruent):

equation

The regular polygon has a center which is also the center of the circle in which it can be inscribed.

Some regular polygons tile the plane, i.e. the plane can be covered by such polygons without gaps or overlaps.

Equilateral triangles and regular hexagons are in this category.

An axis of symmetry for a polygon is a line that divides the figure in two figures that are the mirror image of one another.