Area of Regular Polygon

Area of a Polygon is the space covered inside the boundary of any polygon. Polygons are two- dimensional plane figures with at least three or more sides. It is to be noted that a polygon has a finite number of sides. The number of sides in a polygon determines its name. For example, a pentagon is a polygon that has 5 sides, a hexagon has 6 sides, a heptagon has 7 sides, and so on. Regular polygons are class

Regular Polygons

Regular polygons are polygons whose all sides and angles are equal in measure. Thus, all such polygons which are equiangular and equilateral are called regular polygons. The sum of all external angles of a regular polygon is 360 degrees.

Common examples of regular polygons are square, equilateral triangle, regular pentagon, regular hexagon, etc.

Area of Polygons

The region that is enclosed by a polygon is the measure of its area. Since polygons are plane shapes, their areas refer to the regions occupied by them in 2D planes. The area of all kinds of polygons is expressed in square units- square meters, square centimeters, square inches, square feet, etc.

Example: In the figure below, the shaded region depicts the area of a hexagon in the XY- plane.

Difference between Perimeter and Area of Polygons

The difference between the area and perimeter of Polygons can easily be understood using the table discussed below,

	Perimeter	Area
Meaning	It refers to the length of the boundary of a polygon.	It is the region enclosed between the boundary of the polygon.
Formula	Perimeter = Sum of lengths of all sides	Area of a particular polygon can be calculated using a particular formula.
Unit	It is expressed in units like feet, inches, meters, etc.	Area of all kinds of polygons is expressed in square units- square meters, square centimetres, square inches, square feet, etc.

Area of Regular Polygons

Area of the Regular Polygon is space occupied inside the boundaries of the regular polygon.

Formulae for Areas of Some Specific Regular Polygons

Polygon Formula for Areas of Some Specific Regular Polygons are given in the table below,

Regular Polygon	Formula
Equilateral Triangle
Square	(side)2
Regular Pentagon
Regular Hexagon

 Areas of Regular Polygons

When the length and number of sides of the regular polygon are given then its area is calculated using the formula,

Area = (n × l × apothem) / 2

where,
apothem is l / {2 tan (180^° / n)}
n is the number of sides
l is the length of side

Area of Irregular Polygons

An irregular polygon is a polygon whose sides and interior angles all are of a different measure. Scalene triangles, rectangles, kites, etc. are examples of irregular polygons. Calculation of the Area of Irregular Polygons involves splitting up an irregular polygon into a set of regular polygons so that formulas pertaining to the areas of regular polygons can be used to calculate the area of the given irregular polygon. Study the example given below to get an idea of how the area of an irregular polygon is calculated.

Example: Find the area of the following Irregular Polygon given below,

Solution:

Divide the whole polygon into smaller polygons as follows:

Now the given polygon has been divided into trapezium BCDE and triangle ABE.

Area of trapezium BCDE = 1/2 x Sum of parallel sides x Altitude

= 1/2 x 5 x 10 x 3

= 75 sq. m.

Area of triangle ABE = 1/2 x Base x Height 

= 1/2 x 10 x (9 – 4)

= 1/2 x 10 x 5

= 25 sq. m.

Thus, required area = Area of trapezium BCDE + Area of triangle ABE

= 75 sq. m. + 25 sq. m

= 100 sq. m.

Read, More

• Area of Polygons
• Area of a Triangle
• Area of a Square

Solved Examples on Area of Polygon

Example 1: Find the area of the following polygon,

Solution:

Divide the whole polygon into smaller polygons as seen in the above diagram

Clearly, the polygon obtained by joining the interior vertices of the polygon is a regular pentagon with side 7 units.

Required Area = Area of regular pentagon with side 7 units + 5(Area of equilateral triangle with side 7 units)

= 

= 683.811 sq. units

Example 2: Find the area of the following irregular polygon.

Solution:

Divide the whole polygon into smaller polygons as follows,

The figure gets divided into two rectangles- R1 and R2.

Area of polygon = Area of R₁ + Area of R₂

= 9 x 6 + 5 x 18

= 54 + 90

= 144 square meters.

Example 3: What is the perimeter and area of a regular hexagon whose side is 9 cm?

Solution:

Perimeter of the hexagon = 6 x length of side

                                         = 6 x 9

                                         = 54 cm.

Area of a regular hexagon = 

                                           = 210.45 sq. cm.

Example 4: Find the perimeter and area of a regular nonagon with a side of 10 cm.

Solution:

A regular nonagon has 9 equal sides. 

Given:

Length of each side = 10 cm.

So, Perimeter = n x l = 9 x 10 = 90 cm.

Apothem =  = 13.73 cm

Now, Area = [Perimeter x Apothem]/2

                 = 90 x 13.73/2

                 = 617.85 sq. cm.

Example 5: Find the area of a regular polygon whose each interior angle measures 144°. Assume that each side of this polygon measures 15 units.

Solution:

Since Each Interior Angle = 144°

Each Exterior Angle = 180° – 144° = 36°

Number of sides of the polygon = 360°/Exterior Angle

                                                  = 360°/36°

                                                  = 10 sides

Thus, the given polygon is a decagon.

Apothem =  = 23.08 units

Area = [n x l x Apothem]/2

        = [10 x 15 x 23.08]/2

        = 1731 sq. units

Example 6: Find the area of a rectangular octagon that has been cut from a square of side 8 cm.

Solution:

In order to form an octagon from a square, 4 right triangles need to be cut, each from each of the four corners of the given square. This is shown below:

Now, side of the octagon = 8(√2 – 1) = 3.31 cm

Apothem = 3.99 cm

Area = [n x l x Apothem]/2

         = 8 x 3.31 x 3.99/2

         = 52.87 sq. cm.

FAQs on Area of Polygon

Question 1: What is a regular polygon?

Answer:

A polygon whose all sides and angles are equal in measure is a regular polygon.

Question 2: What do you mean by the area of a polygon?

Answer:

The region that is enclosed by a polygon is the measure of its area. It is expressed in square units.

Question 3: How many equilateral triangles can a regular hexagon be divided into?

Answer:

A regular hexagon is a hexagon with all side equal. It can be divided into six equilateral triangles.

Question 4: What is the area of a polygon with n sides?

Answer:

Area = (n × l × apothem) / 2

where,
apothem is l / {2 tan (180^° / n)}
n is the number of sides
l is the length of side

Question 5: What do you mean by the apothem of a polygon?

Answer:

The line segment joining the center of a polygon to the midpoint of one of its sides is called the apothem of the polygon.