What is a Polygon? Shape, Types, Formulas, Examples, Facts

what is a polygon?

In geometry, a polygon can be defined as a flat or two-dimensional closed shape bounded by straight sides. it has no curved sides. the sides of a polygon are also called its edges. the points where two sides meet are the vertices (or corners) of a polygon.

Here are some examples of polygons.

here are some examples of a polygon

polygon graph

Polygons are named by the number of sides they have. polygons are usually denoted by n-gon where n represents the number of sides it has, for example, a five-sided polygon is called a 5-gon, a ten-sided one is called a 10-gon, and so on.

However, few polygons have special names. the minimum number of sides a polygon can have is 3 because it needs a minimum of 3 sides to be a closed shape or else it will be open.

Although polygons with sides greater than 10 also have special names, we usually denote them with n-gon since the names are complex and not easy to remember.

polygon types

Polygons can be classified based on the number of sides and angles they have:

1. classification according to sides: regular and irregular polygons:

regular polygons: Polygons that have equal sides and equal angles are regular polygons.

For example, an equilateral triangle is a regular polygon with three sides. A square is a regular polygon with four sides. a regular hexagon is a regular polygon with six sides.

here are some examples of regular polygons.

irregular polygons – polygons with unequal sides and angles are irregular polygons.

here are some examples of irregular polygons.

1. classification based on angles: convex and concave polygons:

convex polygons: A convex polygon is a polygon with all interior angles less than 180°.

in convex polygons, all diagonals are inside the polygon.

(diagonal is a line segment joining two non-consecutive vertices of a polygon)

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here are some examples of convex polygons.

Concave polygons: A concave polygon is a polygon with at least one interior angle greater than 180°.

in concave polygons, not all diagonals are inside the polygon.

here are some examples of concave polygons.

difference between convex and concave polygon

3. simple and complex polygon:

simple polygon: A simple polygon has only one boundary. the sides of a simple polygon do not intersect.

complex polygon: complex polygon is a polygon whose sides intersect one or more times.

sum of the angles of a polygon

1. sum of the interior angles of a polygon:

sum of the interior angles of a polygon with n sides = (n – 2) × 180°

for example: consider the following 6-sided polygon

here, ∠a + ∠b + ∠c + ∠d + ∠e + ∠f = (6 – 2) × 180° = 720° (n = 6 since the polygon has 6 sides)

2. sum of the exterior angles of the polygons

sum of the exterior angles of the polygons = 360°

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the sum will always equal 360 degrees, regardless of how many sides it has.

for example: consider the following 5-sided polygon

here, ∠m + ∠n + ∠o + ∠p + ∠q = 360°

angles in regular polygons

in a regular polygon, all its

• sides are equal
• interior angles are equal
• exterior angles are equal

interior angle:

sum of the interior angles of a polygon with n sides = (n – 2) × 180°

so, each interior angle = (n – 2) × 180n

exterior angle:

sum of the exterior angles of the polygons = 360°

so, each exterior angle = 360°n

sum of interior angle and exterior angle:

whether the polygon is regular or irregular, at each vertex of the polygon the sum of one interior angle and one exterior angle is 180°.

solved examples on polygon

example 1: fill in the blank.

1. the name of the three-sided regular polygon is ________________.
2. A regular polygon is a polygon whose _____________ are equal and all angles are equal.
3. The sum of the exterior angles of a polygon is __________.
4. A polygon is a simple closed figure made up of only _______________.

solution:

1. equilateral triangle
2. sides
3. 360°
4. line segments

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Example 2: Write the number of sides of a given polygon.

1. nonagon
2. triangle
3. pentagon
4. decagon

solution:

1. 9
2. 3
3. 5
4. 10

Example 3: Find the measure of each exterior angle of a regular 20-sided polygon.

solution:

the polygon has 20 sides. so, n = 20.

sum of the exterior angles of the polygons = 360°

so, each exterior angle = 360°n = 360°20 = 18°

example 4: the sum of the interior angles of a polygon is 1620°. How many sides does it have?

solution:

sum of the interior angles of a polygon with n sides = (n – 2) × 180°

1620° = (n – 2) × 180°

n – 2 = 1620180

n – 2 = 9

n = 9 + 2

n = 11

so, the given polygon has 11 sides.

practice problems

ffrequently asked questions