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Polygon – Shape, Types, Formulas, Examples

When your kids learn, they will come across geometry and get acquainted with a new set of shapes called the polygons. These sets of shapes often come in simple and complex packages, and kids would need extra help understanding the polygon geometry to get through classes, assignments, and tests easily. Here is all you need to know if you want to help your kid understand basic things about the polygon shapes. 

What is a Polygon?

A polygon definition is simple. A polygon is a flat shape made of straight lines that connect and form a closed shape. Each side has to be a straight line. The number of inside corners of a polygon depends on how many sides it has, and you can figure out how many corners there are by using a math formula. A polygon comes in many exciting shapes.

Identifying Polygons 

To identify a polygon, teachers should help kids seek out a closed geometric figure composed of straight lines. A polygon has at least three sides; each side must have a straight line connecting it to the other side. Identifying a polygon is like connecting dots with straight lines. You must ensure that the shape is not twisted and check if it is crossing over itself. Some examples of polygons are triangles, rectangles, and squares, and there are shapes with many more sides, like the hexagon. To identify a polygon, count the number of sides and check that each side is straight.

Polygon Chart

A polygon chart is a graph that displays information using a circular grid with lines extending out from the center; this chart resembles a spider web. Each line represents a distinct variable, such as temperature or speed, that is being measured. The information is presented as a series of connected dots or lines that creates a polygon shape. The shape of the polygon provides insight into the connections between the variables being measured. Polygon charts are frequently used in fields such as sports and marketing to compare and contrast data.

Types of a Polygon 

There are two major types of polygons: regular and irregular polygons. There are also other types, but the following are the major ones:

Regular Polygon 

A regular polygon has the sides that are all equal in length, and all the interior angles measure the same degrees. The examples of regular polygons include equilateral triangles, squares, and regular pentagons.

Irregular Polygons 

The polygon sides often have different lengths and interior angles in irregular polygons. Rectangles, rhombuses, and irregular pentagons are the examples of irregular polygons. 

Some other polygons include:

• Convex polygons 
• Concave polygons 
• Trigons 
• Quadrilateral polygons 
• Equilateral polygons 
• Equiangular polygons 

Classification based on sides 

Polygons can be classified in different ways, and one of the ways is based on the shape’s sides. Here is how you can classify polygons based on sides: 

• Triangle: A polygon with three sides.  
• Quadrilateral: A polygon with four sides. The examples are squares, rectangles, and parallelograms.
• Pentagon: A polygon with five sides. 
• Hexagon: A polygon with six sides. You can find this at the bottom of pencils and on stop signs. 
• Heptagon: A polygon with seven sides.
• Octagon: A polygon with eight sides.
• Nonagon: A polygon with nine sides.
• Decagon: A polygon with ten sides.
• Undecagon: A polygon with eleven sides.
• Dodecagon: A polygon with twelve sides.

Polygons with more than twelve sides can be named using the Greek numerical prefixes for the number of sides, followed by “-gon.” For example, a polygon with 15 sides is called a pentadecagon. These types can also be called polygon names or names of polygons.

Classification based on angles

Another way to classify polygons is by measuring their interior angles, and here is how you can do that:

Equiangular polygon: A polygon with all angles measuring the same degrees. Examples include squares, equilateral triangles, and regular polygons.

Acute polygon: All angles measure 90 degrees in an acute polygon. 

Right polygon: In a right polygon, only one angle measures precisely 90 degrees, and this is usually the right angle. The examples of right-angle polygons are rectangles and squares.

Obtuse polygon: A polygon with one angle measuring more significant than 90 degrees but less than 180 degrees.

Straight polygon: A polygon with all angles measuring precisely 180 degrees. This type of polygon is not common as it describes a line.

As a side note, if a polygon has one or more angles measuring 180 degrees, it is no longer a polygon but rather a degenerate polygon or a line segment.

Simple and complex polygon 

A simple polygon has no self-intersections or holes, and its sides do not cross. Instead, the sides of a simple polygon only meet at their endpoint, and the polygon encloses a single, connected region. Some examples of simple polygons are triangles, rectangles, and regular polygons. Because of their distinct properties, simple polygons can be analyzed using formulas and theorems.

A complex polygon is a polygon that has self-intersections or holes, and its sides cross each other. This results in multiple disconnected regions within the polygon. The examples of complex polygons include irregular polygons, concave polygons, and star-shaped polygons.

Compared to simple polygons, complex polygons are more challenging to analyze mathematically. For example, computing the area or perimeter of a complex polygon may require breaking it down into simpler shapes, and geometric formulas and theorems may not apply in some cases. However, complex polygons are helpful in fields such as computer graphics and geography to model and represent complex shapes and structures.

Angles of Polygon

The sides of the polygon form the angles of a polygon. The formula to find the total of the angles of a polygon is (n – 2) x 180 degrees, where n is the number of sides. For example, a triangle has three sides, so if n is 3, (3 – 2) x 180 degrees = 180 degrees.

If you have a regular polygon, you can find the measure of each interior angle by using the formula (n – 2) x 180 degrees / n, where n is the number of sides. For example, a hexagon has six sides, so you plug in 6 for n and get (6 – 2) x 180 degrees / 6 = 120 degrees.

However, for concave polygons that have at least one angle greater than 180 degrees, finding the measure of each interior angle is more complex, and you may have to break it into smaller polygons. 

Area and Perimeter Formulas

Area and perimeter are essential measurements used to describe the size and shape of a polygon. The formulas for calculating the area and perimeter of some standard polygons are:

Triangle:

Area = (base x height) / 2

Perimeter = sum of all sides

Rectangle:

Area = length x width

Perimeter = 2 x (length + width)

Square:

Area = side x side

Perimeter = 4 x side

Regular Polygon:

Area = (1/2) x perimeter x apothem, where apothem is the distance from the center of the polygon to the midpoint of a side

Perimeter = number of sides x length of each side

You must ensure that the units for measurement for length and area are the same and must remain consistent; and the apothem value may need to be rounded depending on the level of precision required.

Solved Examples on Polygon

Example one: 

Find the perimeter and area of a rectangle with a length of 8 cm and a width of 6 cm.

Solution:

Perimeter = 2 x (length + width) = 2 x (8 cm + 6 cm) = 28 cm

Area = length x width = 8 cm x 6 cm = 48 cm²

Example two: 

Find the perimeter and area of a parallelogram with a base of 12 cm, height of 8 cm, and the side length of 10 cm.

Solution:

Perimeter = sum of all sides = 2 x (base + side length) = 2 x (12 cm + 10 cm) = 44 cm

Area = base x height = 12 cm x 8 cm = 96 cm²

Example three:

Find the perimeter and area of a regular hexagon with a side length of 4 cm.

Solution:

Perimeter = number of sides x length of each side = 6 x 4 cm = 24 cm

Area = (1/2) x perimeter x apothem = (1/2) x 24 cm x (√3 x side / 2) = 12√3 cm² ≈ 20.8 cm²

Frequently Asked Questions on Polygon

What is the diagonal of a polygon?

The diagonal of a polygon is a straight line segment that connects two non-adjacent vertices of the polygon. 

What are the properties of regular polygons?

All sides and angles are the same.

Regular polygons have rotational symmetry.

The sum of interior angles is (n-2) × 180 degrees, and n is the number of sides.

Each exterior angle measures 360 degrees divided by the number of sides.

The number of diagonals is n × (n-3) divided by 2, where n is the number of sides.

The area can be calculated using the formula (1/2) × perimeter × apothem.

Is a circle a polygon?

No, a circle is not a polygon.

What is the minimum number of sides a polygon must have?

A polygon must have at least three sides. 

Can a number of angles and a number of sides of a polygon be different?

The sides and angles must be the same for a polygon to be defined as one. 

How do you know if a polygon is regular?

A polygon is regular when all the sides and the angles are the same. 

How are polygons named?

Polygons are names based on the number of sides they have.