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A regular polygon, remember, is a polygon whose sides and interior angles are all congruent. To understand the formula for the area of such a polygon, some new vocabulary is necessary.

The center of a regular polygon is the point from which all the vertices are equidistant. The radius of a regular polygon is a segment with one endpoint at the center and the other endpoint at one of the vertices. Thus, there are n radii in an n-sided regular polygon. The center and radius of a regular polygon are the same as the center and radius of a circle circumscribed about that regular polygon.

An apothem of a regular polygon is a segment with one endpoint at the center and the other endpoint at the midpoint of one of the sides. The apothem of a regular polygon is the perpendicular bisector of whichever side on which it has its endpoint. A central angle of a regular polygon is an angle whose vertex is the center and whose rays, or sides, contain the endpoints of a side of the regular polygon. Thus, an n-sided regular polygon has n apothems and n central angles, each of whose measure is 360/n degrees. Every apothem is the angle bisector of the central angle that contains the side to which the apothem extends. Below are pictured these characteristics of a regular polygon.

Once you have mastered these new definitions, the formula for the area of a regular polygon is an easy one. The area of a regular polygon is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter.

When an n-sided polygon is split up into n triangles, its area is equal to the sum of the areas of the triangles. Can you see how 1/2(ap) is equal to the sum of the areas of the triangles that make up a regular polygon? The apothem is equal to the altitude, and the perimeter is equal to the sum of the bases. So 1/2(ap) is only a slightly simpler way to express the sum of the areas of the n triangles that make up an n-sided regular polygon.

## What is the Area of a Regular Polygon?

The area of a regular polygon is the space enclosed by the boundary of the regular polygon. In other words, the area of a regular polygon is the area that is enclosed by it. It is generally measured in square units, such as cm^{2}, m^{2}, ft^{2}, and so on. Generally, the area of a polygon can be determined using different formulas, based on whether the polygon is regular or irregular.

## Area of Regular Polygon Formula

The area of regular polygon formulas for some of the most commonly used polygons are as follows:

**Area of Equilateral triangle = (√3a ^{2}) /4 square units**

Where “a” is the side length of an equilateral triangle

**Area of Square = a ^{2} square units**

Where “a” is the side length of square

**Area of Regular Pentagon = (1/4) ×√[5(5+2√5)] ×a ^{2} square units**

Where “a” is the side length of the pentagon

**Area of Regular Hexagon = [3√3a ^{2}]/2 Square units**

Where “a” is the side length of the hexagon.

## Area of Regular Polygon of N-sides Formula

If “n” is the number of sides of a polygon, then the formula to find the area of regular polygon of n sides is given by:

**Area of Regular Polygon Formula = [l ^{2}n]/[4tan(180/n)] Square units**

Where “l” is the side length of polygon

“n” is the number of sides of the polygon.

## Area of Regular Polygon Inscribed in a Circle

The area of a regular polygon inscribed in a circle formula is given by:

**Area of a regular polygon inscribed in a circle = (nr ^{2}/2) sin (2π/n) square units**

Where “n” is the number of sides

“r” is the circumradius.

A regular polygon is a polygon that has equal sides and equal angles. Thus, the technique to calculate the value of the area of regular polygons is based on the formulas associated with each polygon. Let us have a look at the formulas of some commonly used regular polygons:

Equilateral Triangle | Area = (√3 ×(length of a side)^{2})/4 |

Square | Area = (length)^{2} |

In order to determine the area of a regular polygon, if the number of its sides are known, is given by:

- Area of regular polygon = (number of sides × length of one side × apothem)/2, where the length of apothem is given as the

- , where l is the side length and n is the number of sides of the regular polygon.
- In terms of the perimeter of a regular polygon, the area of a regular polygon is given as, Area = (Perimeter × apothem)/2, in which perimeter = number of sides × length of one side

**Example: **Find the area of a regular pentagon whose side is 7 inches long.**Solution: **Given the length of one side = 7 inches.

Hence, the area of the regular pentagon is given as A =

⇒ A = 84.3 square inches

Thus, the area of the regular pentagon is 84.3 square inches.

### Area of Irregular Polygons

An irregular polygon is a plane closed shape that does not have equal sides and equal angles. Thus, in order to calculate the area of irregular polygons, we split the irregular polygon into a set of regular polygons such that the formulas for their areas are known. Consider the example given below.

The polygon ABCD is an irregular polygon. Thus, we can divide the polygon ABCD into two triangles ABC and ADC. The area of the triangle can be obtained by:

Area of polygon ABCD = Area of triangle ABC + Area of triangle ADC

## Area of Polygons with Coordinates

The area of polygons with coordinates can be found using the following steps:

- Step 1: First we find the distance between all the points using the distance formula,

- Step 2: Once, the dimensions of the polygons are known find whether the given polygon is a regular polygon or not.
- Step 3: If the polygon is a regular polygon we use the formula, area of regular polygon = (number of sides × length of one side × apothem)/2, where the length of apothem is given as the (length of one side)/(2 ×(tan(180°/number of sides))). If the polygon is an irregular polygon, it is to be divided into several regular polygons to find the area.

**Example:** What is the area of the polygon formed by the coordinates A(0,0), B(0, 2), C(2, 2), and D(2, 0)?**Solution: **On plotting the coordinates A(0,0), B(0, 2), C(2, 2), and D(2, 0) on an XY plane and joining the dots we get,

It can be seen, the obtained figure shows a four-sided polygon. In order to understand whether it is a regular polygon or not, we find the distance between all the points.

Now that we know the length of all sides of the given polygon is the same, it shows, it is a square. Thus, the area of the polygon ABCD is given as A = (length)^{2} = (2)^{2} = 4 square units.

Hence, the area of the polygon with coordinates (0,0), (0, 2), (2, 2), and (2, 0) is 4 square units.

**Important Notes**

- If the length of one side is given, it possible to find the area of the regular polygon by finding apothem.
- Apothem falls on the midpoint of a side at the right angle dividing it into two equal parts.
- An Equilateral triangle is a regular polygon with 3 sides, while a square is a regular polygon with 4 sides. Hence, they are not prefixed as regular ahead of the shape name.

## How to Find the Area of Regular Polygons

A regular polygon is a 2-dimensional convex figure with congruent sides and angles equal in measure.^{ }Many polygons, such as quadrilaterals or triangles have simple formulas for finding their areas, but if you’re working with a polygon that has more than four sides, then your best bet may be to use a formula that uses the shape’s apothem^{ }and perimeter. With a little bit of effort, you can find the area of regular polygons in just a few minutes.

### Calculating the Area

**Calculate the perimeter.**The perimeter is the combined length of the outline of any two-dimensional figure. For a regular polygon, it can be calculated by multiplying the length of one side by the number of sides (*n*).

2. **Determine the apothem.** The apothem of a regular polygon is the shortest distance from the center point to one of the sides, creating a right angle. This is a little trickier to calculate than the perimeter.

- The formula for calculating the length of the apothem is this: the length of the side (
*s*) divided by 2 times the tangent (tan) of 180 degrees divided by the number of sides (*n*).

3. **Know the correct formula.** The area of any regular polygon is given by the formula: **Area = ( a x p)/2**, where

**a**is the length of the apothem and

**p**is the perimeter of the polygon.

4. **Plug the values of a and p in the formula and get the area.** As an example, let’s use a hexagon (6 sides) with a side (*s*) length of 10.

- The perimeter is 6 x 10 (
*n*x*s*), equal to 60 (so*p*= 60). - The apothem is calculated by its own formula, by plugging in 6 and 10 for
*n*and*s*. The result of 2tan(180/6) is 1.1547, and then 10 divided by 1.1547 is equal to 8.66. - The area of the polygon is
*Area*=*a*x*p*/ 2, or 8.66 multiplied by 60 divided by 2. The solution is an area of 259.8 units. - Note as well, there are no parenthesis in the “Area” equation, so 8.66 divided by 2 multiplied by 60, will give you the same result, just as 60 divided by 2 multiplied by 8.66 will give you the same result.

### Understanding the Concepts in a Different Way

**Understand that a regular polygon can be thought of as a collection of triangles.**Each side represents the base of a triangle, and there are as many triangles in the polygon as there are sides. Each of the triangles are equal in base length, height, and area.

2. **Remember the formula for the area of a triangle.** The area of any triangle is 1/2 times the length of the base (which, in the polygon, is the length of a side) multiplied by the height (which is the same as the apothem in regular polygon).

3. **See the similarities.** Again, the formula for a regular polygon is 1/2 times the apothem multiplied by the perimeter. The perimeter is just the length of one side multiplied the by the number of sides (*n*); for a regular polygon, *n* also represents the number of triangles that make up the figure. The formula, then, is nothing more than the area of a triangle multiplied by the number of triangles in the polygon.

## Difference Between Perimeter and Area of Polygons

The perimeter and area of polygons are both measurable values that depend on the length of sides of the polygon. In order to differentiate between both of them, it is necessary to understand the basic difference between perimeter and area. Observe the table given below to understand this difference better.

Definition | It is defined as the total length of the boundary of the polygon which can be obtained by adding the length of all its sides. | It is defined as the region or space enclosed by any polygon. |

Formula | The perimeter of Polygon = Length of Side 1 + Length of Side 2 + …+ Length of side N (for an N sided polygon) | The area of polygons can be found by different formulas depending upon whether the polygon is a regular or an irregular polygon. |

Unit | The unit of the perimeter of polygons is expressed in meters, centimeters, inches, feet, etc. | The unit of the area of polygons is expressed in (meters)^{2}, (centimeters)^{2}, (inches)^{2}, (feet)^{2}, etc. |

The similarity between the calculation of perimeter and area of a polygon is that both depend on the length of the sides of the shape and not on the interior angles or the exterior angles of the polygon.

## Area of Polygon Formulas

A polygon can be categorized as a regular or an irregular polygon based on the length of its sides. Thus, this differentiation also brings a difference in the calculation of the area of polygons. The area of some commonly known polygons is given as:

- Area of triangle = (1/2) × base × height

We can also find the area of a triangle if the length of its sides is known by using Heron’s formula which is,

- where s = Perimeter/2 = (a + b + c)/2, a, b, and c are the length of its sides.
- Area of rectangle = length × width
- Area of parallelogram = base × height
- Area of trapezium = (1/2) × (sum of lengths of its parallel sides or bases) × height
- Area of rhombus = (1/2) × (product of diagonals)

In order to calculate the area of a polygon, it must be first known whether the given polygon is a regular polygon or an irregular polygon.

### Area of Regular Polygon Problems and Answers

Go through the below problems to find the area of a regular polygon.

**Example 1:**

Find the area of a regular hexagon whose side length is 2 cm.

**Solution:**

Given: Side length, a = 2 cm

We know that the formula for the area of a regular hexagon is [3√3a^{2}]/2 Square units

Substituting the value in the formula, we get

A = [3√3(2)^{2}]/2

A = (12√3)/2 = 6√3

We know that √3 = 1.732

So, A = 6(1.732) = 10.392 cm^{2}

Therefore, the area of a regular hexagon with a side length of 2 cm is 10.392 cm^{2}.

**Example 2:**

Calculate the area of a regular polygon whose side length is 6 cm and the number of sides is 5.

**Solution:**

Given that, the number of sides,n = 5

Side length, l = 6 cm

We know that,

Area of Regular Polygon Formula = [l^{2}n]/[4tan(180/n)] Square units

Now, substitute the values in the formula, we get

A = [(6^{2}(5)]/[4tan(180/5)] cm^{2}

A = 180/[4tan (180/5)] cm^{2}

A = 180/ [4(tan 36)] cm^{2}

A = 180/[4(0.7265)] cm^{2}

A = 180/2.906 cm^{2}

A = 61.94 cm^{2}

Hence, the area of a regular polygon is 61.94 cm^{2}.

**Example 3:**

Find the area of regular pentagon inscribed in a circle whose circumradius is 4 cm.

**Solution:**

Given: Number of sides, n = 5

Circumradius, r = 4 cm.

We know that the area of a regular polygon inscribed in a circle = (nr^{2}/2) sin (2π/n) square units.

Substituting the values, we get

A = (5(4)^{2}/2) sin (2π/5) cm^{2}

A = (5(16)/2) sin (360/5) cm^{2}

A = 40 sin (72) cm^{2}

A = 40(0.951) cm^{2}

A = 38.04 cm^{2}

Therefore, the area of a regular pentagon inscribed in a circle is 38.04 cm^{2}.

## Area of Polygons Examples

**Example 1: Find the area of the polygon given in the image.**

**Solution:** It can be seen that the given polygon is an irregular polygon. The area of polygon can be found by dividing the given polygon into a trapezium and a triangle where ABCE forms a trapezium while ECD forms a triangle. In order to find the area of polygon let us first list the given values:

For trapezium ABCE,

Length of AB = 5 units

Length of EC = 8 units

Height of the trapezium = 3 units

Thus, the area of the trapezium ABCE = (1/2) × (sum of lengths of bases) × height = (1/2) × (5 + 8) × 3

⇒ Area of trapezium ABCE = (1/2) × 13 × 3 = 19.5 square units

For triangle ECD,

Length of EC = 8 units

Height of triangle = (7 – 3) units = 4 units

Thus, the area of triangle ECD = (1/2) × base × height = (1/2) × 8 × 4

⇒ Area of triangle ECD = (1/2) × 8 × 4 = 16 square units

The area of the polygon ABCDE = Area of trapezium ABCE + Area of triangle ECD = (19.5 + 16) square units = 35.5 square units

∴ The area of the given polygon is 35.5 square units.

**Example 2: Determine the length of the rectangle,** i**f the area of a rectangle is 625 square units and the width (breadth) is 5 units.****Solution: **Given, area of polygon (Rectangle) = 625 square unitsand width of rectangle = 5 units. Thus, the length of the rectangle is calculated as:Area of rectangle = length × breadth

⇒ Area of rectangle = length × 5 = 625

⇒ Length of rectangle = 625/5 = 125 unitsThus, the length of the rectangle is 125 units.

### Area of Regular Polygon Practice Questions

Solve the following problems:

- Find the area of a regular hexagon whose side length is 4 cm.
- Compute the area of a regular polygon of side length 6 cm, whose number of sides is 7.
- Determine the area of a regular pentagon whose side length is 8 cm.
- Find the area of a regular pentagon inscribed in a circle whose circumradius is 4 cm.

**Frequently Asked Questions on Area of a Regular Polygon**

Q1

### What is the area of a regular polygon?

The area of a regular polygon is the space enclosed by the boundary of the regular polygon. The area of a regular polygon is generally measured in square units.

Q2

### What is the formula for the area of a regular polygon of n-sides?

The formula for the area of a regular polygon of n-sides is [l^{2}n]/[4tan(180/n)] Square units, where l is the side length of the polygon and n is the number of sides.

Q3

### What is the formula for the area of a regular polygon inscribed in a circle?

The area of a regular polygon inscribed in a circle formula is (nr^{2}/2) sin (2π/n) square units, where n is the number of sides and r is the circumradius.

Q4

### Does the area of a regular polygon directly depend on the exterior angle?

No, the area of a regular polygon does not directly depend on the exterior angle.

Q5

### What is the difference between the perimeter and the area of a regular polygon?

The area of a regular polygon is the space occupied by it, whereas the perimeter of a regular polygon is the total length of the boundary of a regular polygon.

### What is the Area of a Polygon?

The definition of the area of a polygon is the measure of the area that is enclosed by it. As polygons are closed plane shapes, thus, the area of a polygon is the space that is occupied by it in a two-dimensional plane. The unit of the area of any polygon is always expressed in square units. Observe the following figure which shows the area of a polygon on a two-dimensional plane.

### What is Area of Polygon Definition?

The space enclosed by any polygon in a two-dimensional plane is defined as the area of the polygon. We write the unit of area of a polygon in square units^{ }where the unit can be SI units like meters or centimeters, etc. or USCS units (inches or feet, etc).

### How to Find the Area of Polygons?

The area of a polygon can be calculated by understanding whether the given polygon is a regular polygon or an irregular polygon. The steps to calculate the area of polygons are:

- Step 1: Find whether the given polygon is a regular polygon or not.
- Step 2: If it is a regular polygon or has a standard formula of calculation, use it to determine the value with all the given dimensions of the polygon, otherwise, the area of the polygon can be calculated by dividing it into a set of regular polygons whose area can be added to get the area of the required polygon.

### What is the Area of Polygon with n sides?

If the number of sides of a polygon is given, the area of the polygon can be calculated with the help of the formula, Area = [(L^{2} n)/4 tan(180/n)]; where L is the length of its side and ‘n’ is the number of sides of the polygon.

### What is the Difference Between the Perimeter and Area of Polygons?

The perimeter of a polygon is the total length of its boundary, whereas, the area of a polygon is the region occupied by it. The perimeter of a polygon is obtained by adding the length of all its sides while the area of a polygon is obtained by using the required formula depending upon whether the given polygon is a regular polygon or not. The unit of the perimeter of a polygon is always given in linear units as it is one-dimensional while the unit of the area of a polygon is always given in square units because the area is a two-dimensional concept.

### How do you find the Area of Polygons with Vertices?

The area of polygons with vertices can be found using the following steps:

- Step 1: Find the distance between all the points using the distance formula,

- Step 2: Once we know the dimensions of the polygons are known we determine whether the given polygon is a regular polygon or not.
- Step 3: If the conclusion from Step 2 shows the polygon is a regular polygon we use the formula, area of regular polygon = (number of sides × length of one side × apothem)/2, where the length of apothem is given as, Apothem = [(length of one side)/{2 ×(tan(180/number of sides))}]. On the other hand, if the polygon is an irregular polygon, it is divided into several smaller regular polygons by finding the dimension of diagonals using the distance formula.

### How to Find the Area of Irregular Polygons?

In order to calculate the value of the area of an irregular polygon we use the following steps:

- Step 1: Divide the given polygon into smaller sections forming different regular or known polygons.
- Step 2: Find the area of each section individually.
- Step 3: Add the area of each section to obtain the area of the given irregular polygon.

### How to Find the Area of Polygons with Perimeter?

If the perimeter of a polygon is given, then its area can be calculated using the formula: Area = (Perimeter × apothem)/2. In this formula, the apothem should also be known or it can be calculated with the help of the formula, Apothem = [(length of one side)/{2 ×(tan(180/number of sides))}]. Substituting the value of the apothem and the perimeter, the area of the polygon can be calculated.

### What is the Formula to Calculate the Area of Regular Polygons?

The formula to calculate the area of a regular polygon is, Area = (number of sides × length of one side × apothem)/2, where the value of apothem can be calculated using the formula, Apothem = [(length of one side)/{2 ×(tan(180/number of sides))}].

### What is the Area of Polygon Definition?

**Ans.** The area of a polygon is the measurement of the area it encloses. Because polygons are closed-plane shapes, their area is the space they occupy in a two-dimensional plane. The area of any polygon is always measured in square units.

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