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Geometry: Polygons





 



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A polygon is a geometric figure in a two dimensional plane. It can be thought of as a union of line segments that are connected at their endpoints. Also, no two line segments are collinear. Any pair of line segments create angles in the polygon. In fact, the word polygon means “many angled.” The term polygon can refer to either the union of line segments, the bounded interior region or both depending on the context.

A polygon consists of non-collinear line segments connected at the endpoints that enclose a region in a plane.

The line segments of a polygon are referred to as its sides. Also, the line segments or sides can only intersect at the endpoints of other line segments. The endpoints of the line segments that form the sides of the polygon are also called vertices. A single endpoint connection is called a vertex. Interior line segments that connect the endpoints are diagonals and the sum of the lengths of each line segment is the perimeter. The line segments that form the sides of a polygon can also be called edges.

The line segments that connect non-adjacent vertices in a polygon are called diagonals.

A geometric figure with an open side that is missing a line segment and not closed all the way around its perimeter is not a polygon.

Polygons Are Closed

Another property of polygons is that they are closed. This means that there is a continuous path around the sides of the polygon that is not broken or terminated.

A polygon is a geometric figure with a closed path along the line segments that make up its sides. An open figure appears as a polygon with a side missing.

Convex and Concave Polygons

Polygons fall into many types of categories two of these include convex and concave. A convex polygon is one where any point inside can be connected by a line segment such that every point on the line segment is included in the polygon or its interior. Any polygon that does not fit this description is a concave polygon.

A convex polygon is more compact than a concave polygon. All the line segments that connect the vertices of the convex polygon (diagonals) cross the interior of the polygon.

A concave polygon may extend out from the rest of the polygon. In a concave polygon, some the line segments that connect the vertices of the concave polygon (diagonals) cross the exterior of the polygon.

Another way to tell if a polygon is convex or concave is to extend the line segments of the polygon. If the extended line segments intersect at any place other than the endpoints of the line segments and if the interior of the polygon is on both sides of an extended line segment then the polygon is concave.

The extended sides of a convex polygon do not intersect and cross the interior of the convex polygon. The extended sides of a concave polygon can intersect and cross the interior of the concave polygon.

Regular Polygons

If the sides of a polygon all have the same length and if the angles all have the same measure then the polygon can be considered as a regular polygon. In addition, a regular polygon is convex.

A regular polygon has congruent sides and the interior angles are also congruent. An irregular polygon has at least one side or angle incongruent with at least one of the others.

A regular polygon can also be inscribed and circumscribed with circles. When a circle is inscribed in a regular polygon the circle touches all the sides.

An inscribed circle is tangent at the midpoints of the sides of a regular polygon.

When a circle is circumscribed around the outside of a regular polygon, it touches all the vertices.

A circumscribed circle touches all the vertices of a regular polygon.

Number of Sides of a Polygon

A polygon can have any number of sides but there are special names for polygons with a certain number of sides. A Greek prefix is used to represent the number of angles while the root is “gon.” For example, tri means three in Greek and when attached to gon it makes trigon which is a triangle. Three and four sided polygons are usually called triangles and quadrilaterals, respectively.

Polygon DiagramName of PolygonNumber of Sides
The triangle is a regular polygon with 3 sides.Triangle (Trigon)3
The quadrilateral is a regular polygon with 4 sides.Quadrilateral (Quadrigon)4
The pentagon is a regular polygon that has 5 sides.Pentagon5
The hexagon is a regular polygon with 6 sides.Hexagon6
The heptagon is a regular polygon that has 7 sides.Heptagon7
The octagon is a regular polygon with 8 sides.Octagon8
The nonagon is a regular polygon having 9 sides.Nonagon9
The decagon is a regular polygon with 10 sides.Decagon10

The nomenclature of a polygon can be generalized to represent any number of sides. Any number of sides is represented by n and its corresponding polygon is called an n-gon. As a result, a four sided polygon can also be called a 4-gon and a seven sided polygon can also be called a 7-gon.

Polygon Angles

Interior Angles

The sum of the interior angles in a polygon can be found by the following formula:

Angle Sum = (n-2)*180

where n is equal to the number of angles.

Exterior Angles

The sum of the exterior angles of a convex polygon is 360°.

The exterior angles of a convex polygon add up to 360 degrees total.

Polygon Diagonals

The formula for calculating the number of diagonals in a polygon can be stated as follows:

N = n(n-3)/2

where N = number of diagonals and n = number of sides.

The number of diagonals in a polygon is related to the number of sides.

Polygon Center

The center of a regular polygon is the point in the interior that is at an equal distance from the vertices or equidistant from corresponding points in the sides such as the midpoints. The radius is the distance from the center to the vertex of a regular polygon and is equal to the radius of the circumscribed circle.

Apothem

The apothem is a line segment that connects the center of a regular polygon with the midpoint of one of the sides and is perpendicular to the side it bisects. The apothem length is the distance from the center to one of the sides of a regular polygon. It also corresponds to the radius of the inscribed circle inside the regular polygon.

The radius of a regular polygon extends from the center of the polygon to a vertex of the polygon while the apothem extends from the center to one of the sides.

Polygon Area

The area of a regular polygon can be calculated by the following formula:

A = 1/2*aP

Where A = area, a = the length of the apothem and P is the total length of the perimeter which is the sum of all the sides.

The area of a regular polygon is equal to one half times the apothem length times the perimeter length.

Special formulas also exist for geometric figures such as triangles, quadrilaterals and other polygons.

 




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