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Polygon

A polygon is a shape of plane geometry. It’s made of straight lines connected to each other forming a “closed” shape.

Polygon comes from the Greek words ”poly”, meaning “many”, and “gonia”, meaning "angles". So it’s a shape with many angles.

Types of polygons

There are several types of polygons.

Here are the most typical of them:

ConvexConcave
Example of a convex polygon
Example of a concave polygon
All angles are less than 180˚One or more angles are
more than 180˚.

RegularIrregular
Example of a regular polygon
Example of a irregular polygon
All angles and sides are
equal.
Not all angles and sides are
equal.

Convex polygons

Angle sum

The angle sum of a convex polygon with n sides is given by:

Sigmaalpha=(n-2)*180^0

Triangles and rectangles are typical examples of convex polygons.

Let’s try to calculate the angle sum of the these two shapes with the formula above:

Triangle:

Sigmaalpha=(n-2)*180^0=(3-2)*180^0=180^0

Rectangle:

Sigmaalpha=(n-2)*180^0=(4-2)*180^0=360^0

Regular polygons

Regular polygons are the only polygons of which it’s possible to make general formulas applicable to all of them.

The circumference of a regular polygon

C=n*b

where C is the circumference, n is the number of sides, and b is the side length.

The area of a regular polygon

A=frac{1}{4}*n*b^2*cot(frac{pi}{n})

Where A is the area, n is the number of sides, and b is the side length.

Use the following formula instead, if your calculator hasn't got the cotangent feature.

cot(a)=frac{cos(a)}{sin(a)}

Area and circumference of a regular polygon

Number of vertices (n): 
Side length (b): 
Pi: 

Names of polygons

A polygon is named after its number of vertices.

NamemeaningAngle sumExamples
TrigonTriangle180The triangle is an important shape in triangulation.
Triangulation was used in the old days to
determine locations and to create maps.
TetragonQuadrilateral360A football field
Pentagon5-sided shape540The headquarters of the United States
Department of Defense
Hexagon6-sided shape720The hexagon is common in nature and in chemistry,
for example the pattern you see inside a beehive
or the structure of sugar molecules.
Heptagon or septagon7-sided shape900British coins: 20 and 50 pence.
Octagon8-sided shape1080Stop sign in traffic.