## A Polygon Shape - Types and Example

A polygon shape is any geometric shape that is classified by its number of sides and is enclosed by a number of straight sides. However, a polygon is considered regular when each of its sides measures equal in length. For example, a 3-sided polygon is a triangle, an 8 -sided polygon is an octagon, while an 11-sided polygon is called 11-gon or hendecagon. The number of sides of a regular polygon can be computed with the help of interior and exterior angles.

For an Instance, A hexagon is a six-sided polygon, while a triangle is a three-sided one. The internal and exterior angles, which are the inside and outside angles formed by the connecting sides of the polygon, may be used to compute the number of sides of a regular polygon.

Subtract the inside angle from 180 to get the outside angle. If the inner angle was 165, for example, subtracting that from 180 would give you 15.

Divide 360 by the angle difference and 180 degrees. 360 divided by 15 = 24, which is the number of sides of the polygon in this case.

To calculate the number of sides of the polygon, divide 360 by the amount of the exterior angle. For example, if the exterior angle is 60 degrees, then dividing 360 by 60 equals 6, which is the number of sides the polygon has.

### Names of Polygons

### Convex and Concave Polygons

A convex polygon closes in an interior space without appearing "dented." None of the interior angles points inward. In geometrical math, you could have a 4-sided polygon that points outward in all directions, like a kite, or you could have similar four sides so two of them point inward, creating a dart. The dart is concave and the kite is convex.

Each interior angle of a convex polygon measures less than 180°. A concave polygon has a minimum of one angle greater than 180°. Imagine a bowtie-shaped hexagon (6 sides). It will consist of two interior angles greater than 180°.

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### Simple and Complex Polygons

Simple polygons contain no self-intersecting sides. Complex polygons, also known as self-intersecting polygons, contain sides that cross over each other. An example of a complex polygon is a classic star. Most people can sketch a star on a sheet of paper very quickly, but some people label it a pentagram, complex polygon, or self-bisecting polygon.

The family of complex star-shaped polygons usually share the Greek number prefix and use the suffix -gram: pentagram, hexagram, heptagram, octagram, and so on.

### Degrees in Polygons: How to Find Them

A polygon is a two-dimensional object that is closed and made up of three or more linked line segments. Polygons include triangles, trapezoids, and octagons, to name a few. Polygons are often classed based on the number of sides and the angles and sides' relative sizes. Polygons can also be classed as regular or non-regular. Regular polygons have equal-length sides and equal-degree angles. In regular polygons, the degrees of the angles can be calculated, but this is not necessarily the case with non-regular polygons.

The polygon's sides are multiplied by the number of sides. The total of all the inner angle degrees is (n - 2) 180. Subtract 2 from the number of sides and multiply by 180 in this calculation). The total of degrees for an octagon, for example, is (8-2) 180. This adds up to 1,080.

Divide the total calculated in Step 1 by the number of sides if the polygon is regular (all sides and angles are equal). This is the degree of each of the polygon's angles. In a normal octagon, for example, each angle has a degree of 135:1. Divide 1,080 by eight to get the answer.

To determine the exterior angle measure of a regular polygon, multiply the angle from Step 2 by 180 minus the degree. This is the degree of each polygon's outer angle. Because the angle in this example is 135 degrees, the value of the supplementary angle is 45 degrees.

### Solved Examples on a Polygon Shape

Example:

1. Find out the Interior Angle of a Regular Octagon?

Solution:

A regular octagon consists of 8 sides, thus:

Exterior Angle of an octagon =\[ \frac{360°}{8} = 45° \]

Interior Angle of an octagon = 180° − 45° = 135°

Or we could also use the formula to determine the interior Angle of an octagon:

Interior Angle = \[ \frac{(n−2) \times 180° }{n} \]

= \[ \frac{(8−2) \times 180° }{8} \]

= \[ \frac{6 \times 180° }{8} \]

= 135°

Thus, the Interior Angle of an octagon measures 135°

Example:

2. Determine the Interior and Exterior Angles of a Regular Hexagon?

Solution:

A regular hexagon consists of 6 sides, thus:

Exterior Angle of a hexagon= \[\frac{360°}{6} = 60°\]

Interior Angle of a hexagon = 180° − 60° = 120°

### Fun Facts

Interior and exterior angles of a polygon are respectively, the inside and outside angles formed by the connecting sides of the polygon.

To be a polygon, the shape must be flat, circumscribed in space, and be created using only straight sides.

Polygons with congruent angles and sides are regular; while all others are irregular.

Polygons with all interior angles measuring less than 180° are convex

Polygon having a minimum of one interior angle greater than 180° is concave.

Simple polygons don’t cross their sides

Complex polygons have self-bisecting sides.

You can spot Polygons all around you!

### Conclusion

Polygons can be studied and categorized in different ways. You can see that polygons can be regular or irregular, concave or convex, and simple or complex. When you come across an unfamiliar polygon, you can simply identify its properties and classify it correctly.

1. Define Polygon ?

A polygon shape is any geometric shape with a number of straight sides and is characterised by its number of sides. A polygon, on the other hand, is termed regular when each of its sides is the same length. A 3-sided polygon is termed a triangle, an 8-sided polygon is called an octagon, and an 11-sided polygon is called a hendecagon. Interior and exterior angles can be used to calculate the number of sides of a regular polygon.

2. What are a Polygon's Characteristics?

The following are the identifying qualities of any polygon:

a two-dimensional figure

In a space, closing (having an interior and exterior angle)

Straight-sided composition

The sum of all the quadrangles' internal angles is 360 degrees.

It's concave if at least one of the inner angles is bigger than 180 degrees.

A simple polygon is defined as one that does not cross over itself and has only one boundary. Aside from that, it's a complicated polygon.

3. How Do We Work Out a Polygon's Angles?

The answer is to multiply the number of sides of a polygon.

Step 1- The sum of all the inner angle degrees is (n - 2) 180. Subtract 2 from the number of sides of a polygon and multiply by 180 with this formula. The sum of degrees for an octagon, for example, will be (8-2) 180. This adds up to 1,080.

Step 2- Divide the amount calculated in Step 1 by the number of sides in a regular polygon (all sides and angles are equal). In a polygon, this is the degree of each angle.

Example - In a normal octagon, for example, each angle has a degree of 135 degrees: Divide 1,080 by 8 to get the answer. To find the exterior angle of a regular polygon, multiply the angle from Step 2 by the supplement (180 minus the degree).

4. What are the different kinds of polygons?

Let's take a closer look at the vast assortment of polygonal forms available.

A Regular Polygon is made up of congruent sides and angles on the inside. The sides and interior angles of an irregular polygon are not congruent.

A Convex Polygon has no interior angles higher than 180 degrees (it consists of no inward-pointing sides).

A Concave Polygon has one internal angle that is more than 180 degrees.

A Simple Polygon contains no self-bisecting sides and circumscribes a single internal space (boundary). It has self-bisecting sides!

Complex Polygons: it has self-bisecting sides!

5. What are the important formulas for a polygon?

The following are the most important polygon formulas:

A polygon with "n" sides has an interior angle sum of 180°. (n-2)

\[\frac{n(n-3)}{2}\] is the number of diagonals in a "n-sided" polygon.

A regular n-sided polygon's internal angles are measured as \[\frac{(n-2)180°}{n}\].

A regular n-sided polygon's outer angles are measured as \[\frac{360°}{n}\].

6. What are the Properties of a Polygon?

The identifying properties of any polygon are the following:

A 2-dimensional shape

Closing in a space (having an interior and exterior angle)

Composed with straight sides

7. What are the Types of Polygons?

Let's take a look at the huge array of shapes that are polygons and go into detail.

A Regular Polygon: it consists of congruent sides and interior angles.

An Irregular Polygon: it does not have congruent sides and interior angles.

A Convex Polygon: it contains no interior angle greater than 180° (it consists of no inward-pointing sides).

A Concave Polygon: it contains one interior angle greater than 180°.

A Simple Polygon: circumscribes a single interior space (boundary) and does not contain self-bisecting sides.

Complex Polygons: it has self-bisecting sides!