**Polygon Formula:** A polygon is a closed curve or figure made by line segments in which no two line segments cross except at their endpoints and no two line segments with the same endpoint are coincident. A polygon, in other terms, is a basic closed curve made up entirely of line segments. A regular polygon is one with equal sides and angles on all sides. As a result, a regular polygon is equiangular as well as equilateral.

This article will go through the formulas for calculating the areas and perimeters of various polygons, as well as the method for calculating the number of polygon diagonals.

**Learn Concepts on Area of a Polygon**

Table of Contents

- What is a Polygon?
- Types of Polygons
- Convex Polygon
- Concave Polygon
- Regular Polygon
- Irregular Polygons
- Regular Polygon Formula
- Formula for the Number of Diagonals in a Polygon
- Diagonal Formula for Different Types of Polygon
- Perimeter Formula of a Regular Polygon
- Formula to Find the Area of a Polygon
- Solved Examples – Polygon Formula
- Summary
- Frequently Asked Questions (FAQ) – Polygon Formula

## What is a Polygon?

A simple closed curve made up of only line segments is called a polygon. Each straight line in a polygon is called its side. A triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon are called a polygon accordingly as it contains \(3,\,4,\,5,\,6,\,7,\,8,\,9,\,10\) sides, respectively.

Adjacent sides of a polygon are the ones whose two sides have a common end-point (vertex). Adjacent vertices are the ones that are the end-points of the same side of a polygon. Diagonals are the line segment obtained by joining vertices that are not adjacent.

### Types of Polygons

There are four types of polygons, namely:

- Convex Polygon
- Concave Polygon
- Regular Polygon
- Irregular Polygon

### Convex Polygon

A polygon in which each angle is less than \({\rm{18}}{{\rm{0}}^{\rm{o}}}\) is called a convex polygon.

In the above figure, \(PQRS\) is a convex polygon.

### Concave Polygon

A polygon in which at least one angle is more than \({\rm{18}}{{\rm{0}}^{\rm{o}}}\) is called a concave polygon.

In the given figure, \(ABCD\) is a concave polygon. \(\angle BCD\) is more than \({\rm{18}}{{\rm{0}}^{\rm{o}}}\), as shown.

### Regular Polygon

A polygon with all sides and all angles equal is called a regular polygon. An equilateral triangle and a square are well-known examples of a regular polygon.

### Irregular Polygons

Polygons that are not regular are called irregular polygons. In other words, a polygon in which the sides and angles vary are irregular polygons.

A rectangle and a rhombus are examples of an irregular polygon.

### Regular Polygon Formula

For a regular polygon of \(n\) sides, we have:

- Each exterior angle \( = \frac{{{{360}^{\rm{o}}}}}{n}\)
- Each interior angle \( = {180^{\rm{o}}} – {\rm{(each}}\,{\rm{exterior}}\,{\rm{angle)}}\)

\( = {180^{\rm{o}}} – \frac{{36{{\rm{0}}^{\rm{o}}}}}{n}\)

\( = \frac{{n \times {{180}^{\rm{o}}} – \left( {2 \times {{180}^{\rm{o}}}} \right)}}{n}\)

\( = \frac{{(n – 2) \times {{180}^{\rm{o}}}}}{n}\)

Therefore, interior angle \( = \frac{{(n – 2) \times {{180}^{\rm{o}}}}}{n}\)

### Formula for the Number of Diagonals in a Polygon

A diagonal of a polygon is a line segment acquired by joining any two opposite angles or non-adjacent vertices. Based upon the polygon type, based on the number of edges, the number of diagonals and their properties would vary.

If the number of sides of the polygon is \(n\) then the number of diagonals that can be drawn is given by \(\frac{{n(n – 3)}}{2}\).

This formula can be used to compute the number of diagonals in a polygon. It differs based on the type of polygon, based on the number of sides. We can use this formula to calculate the number of diagonals of any polygon without drawing them.

### Diagonal Formula for Different Types of Polygon

- The formula to find the length of the diagonal of a square with the length of a side \(a\) is given by \(d = \sqrt 2 a\)
- The diagonal formula of a rectangle with length \(l\) and breadth \(b\) is given by \(d = \sqrt {{l^2} + {b^2}} \)
- The diagonal length of the parallelogram with angles \(A\) & \(B\) is calculated using the formula:

Diagonal \({d_1} = p = \sqrt {{a^2} + {b^2} – 2ab\,\cos \,A} = \sqrt {{a^2} + {b^2} + 2ab\,\cos \,B} \)

Diagonal \({d_2} = q = \sqrt {{a^2} + {b^2} + 2ab\,\cos \,A} = \sqrt {{a^2} + {b^2} – 2ab\,\cos \,B} \)

\( \Rightarrow {p^2} + {q^2} = 2\left( {{a^2} + {b^2}} \right)\) - If \(p\) and \(q\) are the lengths of the diagonal of a rhombus, then the diagonal formula of a rhombus is given by \(p = 2\frac{A}{{{q^\prime }}}\),
- If \(a\) is the side of a regular pentagon, then the diagonal formula of a regular pentagon is given by \(d = \frac{{1 + \sqrt 5 }}{2}a\).

### Perimeter Formula of a Regular Polygon

The computation of the length of the boundary of any closed figure is known as its perimeter. The perimeter of a regular polygon will be the sum of the lengths of its sides.

If \(a\) represents the side of a regular polygon, then the perimeter of each regular polygon will be as follows.

(i) Perimeter of an equilateral triangle \( = 3a\)

(ii) Perimeter of a square \( = 4a\)

(iii) Perimeter of a rectangle with length \(l\) and breadth \(b\) is given by \(P = 2(l + b)\)

(iv) Perimeter of a regular pentagon \( = 5a\)

(v) Perimeter of a regular hexagon \( = 6a\)

(vi) Perimeter of a regular heptagon \( = 7a\)

(vii) Perimeter of a regular octagon \( = 8a\)

(viii) Perimeter of a regular nonagon \( = 9a\)

(ix) Perimeter of a regular decagon \( = 10a\)

### Formula to Find the Area of a Polygon

Below are the formulas to find the area of the different types of polygons.

- If \(b\) is the base and \(h\) is the height of a triangle, then the area of a triangle \( = \frac{1}{2} \times b \times h\)
- If \(a\) is the length of the side of an equilateral triangle, then the area of an equilateral triangle \( = \frac{{\sqrt 3 }}{4}{a^2}\)
- If \(a\) is the length of the side of a square, then the area of a square \( = {a^2}\)
- If \(l\) is the length and \(b\) is the breadth of a rectangle, then, the area of a rectangle \( = l \times b\)
- The area of a quadrilateral is given by \(\frac{1}{2} \times {\rm{diagonal}} \times {\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{heights}}\)
- The area of a regular pentagon with \(a\) as the length of the side is given by \(A = \frac{1}{4}\sqrt {5(5 + 2\sqrt 5 )} {a^2}\)
- The area of a regular hexagon with \(a\) as the length of the side is given by \(A = \frac{{3\sqrt 3 }}{2}{a^2}\)
- The area of regular heptagon with \(a\) as the length of the side is given by \(A = \frac{7}{4}{a^2}\,\cot \,\frac{\pi }{7}\)
- The area of regular hectagon with \(a\) as the length of the side is given by \(A = 2{a^2}(1 + \sqrt 2 )\)
- The area of regular nonagon with \(a\) as the length of the side is given by \(A = \frac{9}{4}{a^2}\,\cot \,\frac{\pi }{9}\)
- The area of regular decagon with \(a\) as the length of the side is given by \(A = \frac{5}{2}{a^2}\sqrt {5 + 2\sqrt 5 } \)

Also, the general formula to find the area of a regular polygon is given by \(A = \frac{{{l^2}n}}{{4\tan \left( {\frac{\pi }{n}} \right)}}\)

Where, \(l = \) length of the side

\(n = \) number of sides.

### Solved Examples – Polygon Formula

** Q.1. An athlete takes \(10\) rounds of a rectangular park, \(60\;\,{\rm{m}}\) long and \(35\;\,{\rm{m}}\) wide. Find the total distance covered by the athlete.** Given: Length of the rectangular park \( = 60\;\,{\rm{m}}\)

Ans:

The breadth of the rectangular park \( = 35\;\,{\rm{m}}\)

The total distance covered by him in one round will be the perimeter of the park.

Now, the perimeter of rectangular park \( = 2(l + b)\)

\( = 2(60 + 35)\,{\rm{m}}\)

\( = 2(95)\,{\rm{m}} = 190\;\,{\rm{m}}\)

So, the distance covered by the athlete in one round is \(190\;\,{\rm{m}}\).

Therefore, distance covered by the athlete in \(10\) rounds \( = 10 \times 190\;\,{\rm{m}} = 1900\,\;{\rm{m}}\)

So, the total distance covered by the athlete is \(1900\,\;{\rm{m}}\).

* Q.2. A door frame of dimensions \({\rm{4}}\,{\rm{m \times 3}}\,{\rm{m}}\) is fixed on the wall of dimensions \(10\;\,{\rm{m}} \times 10\;\,{\rm{m}}\). Find the total labour charge for painting the wall if the labour charges for painting \(1\;{{\rm{m}}^2}\) of the wall is ₹ \(2.50\).* Painting of the wall has to be done, excluding the area of the door.

Ans:

Area of the door \( = l \times b\)

\( = 4 \times 3 = 12\;{{\rm{m}}^2}\)

Area of wall including door \( = 10\;{\rm{m}} \times 10\;{\rm{m}} = 100\;{{\rm{m}}^2}\)

Area of wall excluding door \( = (100 – 12){{\rm{m}}^2} = 88\;{{\rm{m}}^2}\)

Total labour charges for painting the wall \( =₹ 2.50 \times 88 =₹ 220\)

* Q.3. Find the base of a triangle, if the area of a triangle is \(49\;{\rm{c}}{{\rm{m}}^2}\) and height is \({\rm{4}}\,{\rm{cm}}\).* Given: Area of a triangle \( = 49\;{\rm{c}}{{\rm{m}}^2}\)

Ans:

Height \(h = 4\;{\rm{cm}}\).

We know that, area of a triangle \( = \frac{1}{2} \times b \times h\)

\( \Rightarrow 49 = \frac{1}{2} \times b \times 4\)

\( \Rightarrow b = \frac{{98}}{4}\;{\rm{cm}}\)

\( \Rightarrow b = 24.5\;{\rm{cm}}\)

Therefore, the base of a triangle is \(24.5\;{\rm{cm}}\).

** Q.4. The perimeter of a regular hexagon is \(24\;{\rm{cm}}\). How long is its one side?** Perimeter of a regular hexagon \( = 24\;{\rm{cm}}\)

Ans:

A regular hexagon has \(6\) sides, so we can divide the perimeter by \(6\) to get the length of one side.

One side of hexagon \( = 24\;{\rm{cm}} \div 6 = 4\;{\rm{cm}}\)

Therefore, the length of each side of the regular hexagon is \(4\;{\rm{cm}}\)

** Q.5. What is the area of the regular pentagon with side \(3\;{\rm{cm}}\)?** We know that the area of a regular pentagon with \(a\) as the length of the side is given by \(A = \frac{1}{4}\sqrt {5(5 + 2\sqrt 5 )} {a^2}\).

Ans:

Therefore, \(A = \frac{1}{4}\sqrt {5(5 + 2\sqrt 5 )} \times {3^2}\)

\( = \frac{1}{4}\sqrt {5(5 + 2\sqrt 5 )} \times 9\)

\( = 15.484\;{\rm{c}}{{\rm{m}}^2}\)

Therefore, the area of a regular pentagon with a side of \(3\;{\rm{cm}}\) is \(15.484\;{\rm{c}}{{\rm{m}}^2}\).

### Summary

A polygon is a closed two-dimensional object with straight line segments in mathematics. It’s a two-dimensional form, not a three-dimensional one. There are no curved surfaces in a polygon. At least three sides are required for a polygon. Each line segment’s side must only collide with another line segment at its endpoint. We can simply recognise the form of a polygon-based on its number of sides.

In the above article, we have learned the definition of a polygon, types of the polygon, the formula to find the perimeter and area of different regular polygons and the formula to find the number of diagonals of a regular polygon. Furthermore, we learned the formula to find the length of diagonals of some regular polygons and solved some example problems.

### Frequently Asked Questions (FAQ) – Polygon Formula

*Q.1. How to find the interior angles of a polygon?** Ans:* The formula to find the interior angle of a regular polygon with \(n\) number of sides is given by,

Interior angle \( = \frac{{(n – 2) \times {{180}^{\rm{o}}}}}{n}\)

*Q.2. How to find the area of a polygon formula?** Ans:* The formula to find the area of a regular polygon is given by,

\(A = \frac{{{l^2}n}}{{4\tan \left( {\frac{\pi }{n}} \right)}}\)

Where, \(l = \) length of the side

\(n = \) number of sides.

*Q.3. Can we divide irregular polygon using area formulas?** Ans:* Yes, irregular polygons are polygons that don’t have equal sides or equal angles. When calculating the area of an irregular polygon, divide the polygon into smaller regular polygon shapes. We can then combine those measurements into their respective area formulas and multiply to find the area of one part of the shape.

*Q.4. What is the formula to find the number of diagonals in a polygon?** Ans:* If the number of sides of the polygon is \(n\) then the number of diagonals that can be drawn is given by \(\frac{{n(n – 3)}}{2}\)

*Q.5. What is the formula to find the area of regular heptagon?** Ans:* The area of regular Heptagon with \(a\) as the length of the side is given by \(A = \frac{7}{4}{a^2}\cot \frac{\pi }{7}\)

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