Regular Polygon: Ancient Greeks were aware of polygonal patterns, which appeared in rock formations as early as the 7th century B.C. Polygons also appear in rock formations, usually as flat faces of crystals. A polygon is basically a two-dimensional geometric figure with a finite number of sides. An edge of a polygon consists of straight line segments that are connected end-to-end to form a closed shape. An angle is formed when two lines meet at the vertex or corner. A Regular Polygon is a polygon whose all sides and angles are equal. Thus, a Regular Polygon is both equiangular and equilateral.

In this article we will discuss the different examples of Regular Polygons and their properties and solve problems based on Regular Polygons.

Regular Polygon: Definition

A polygon which is having all sides equal and all angles equal is called a regular polygon. Thus, a regular polygon is both equiangular and equilateral.
Regular polygons are convex in which all the interior angles measure less than \({180^ \circ }.\)

Regular Polygon Formula

For a regular polygon of \(n\) sides, we have:
(i) Each exterior angle \( = \frac{{{{360}^ \circ }}}{n}\)
(ii) Each interior angle \( = {180^ \circ } – \) (each exterior angle)
\( = {180^ \circ } – \frac{{{{360}^ \circ }}}{n}\)
\( = \frac{{n \times {{180}^ \circ } – \left({2 \times {{180}^ \circ }} \right)}}{n}\)
\( = \frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}\)
Therefore, interior angle \( = \frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}\)

Examples of Regular Polygons

The best examples of a regular polygon with three sides and four sides are an equilateral triangle and a square.

The terms equilateral triangle and square refer to the regular \(3 – \)sided and \(4 – \)sided polygons, respectively. The words for polygons with \(n > 4\) sides, for example, pentagon, hexagon, heptagon etc., can refer to either regular or non-regular polygons. However, the terms generally refer to regular polygons in the absence of specific wording.

Equilateral triangle: A triangle with all sides and all angles equal is an equilateral triangle.

Interior angle of an equilateral triangle \( = \frac{{\left({3 – 2} \right) \times {{180}^ \circ }}}{3} = {60^ \circ }\)
The exterior angle of an equilateral triangle \( = \frac{{{{360}^ \circ }}}{3} = {120^ \circ }\)
Therefore, the interior angles of an equilateral triangle are \({60^ \circ }.\)
Square: A square is a plane figure with four equal straight sides and four equal angles.
Interior angle of a square \( = \frac{{\left({4 – 2} \right) \times {{180}^ \circ }}}{4} = {90^ \circ }.\)
The exterior angle of a square \( = \frac{{{{360}^ \circ }}}{4} = {90^ \circ }\)
Therefore, the sides of a square are perpendicular to each other.

Polygon: Definition

A simple closed curve made up of only line segments is called a polygon.

Each straight line of a polygon is called its side.

A polygon is called a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon and decagon, accordingly as it contains \({\text{3,4,5,6,7,8,9}}\) and \(10\) sides, respectively.

The different parts of a polygon are side, vertex, diagonal, interior angle and exterior angle.

Vertex: A point where two sides meet are called a vertex.

Diagonal: A straight line connecting two non-adjacent vertices is called a diagonal, which is not a side.

Interior Angle: Angle formed by two adjacent sides inside the polygon is known as the interior angle.

Exterior Angle: The angle formed by two adjacent sides outside the polygon is known as the exterior angle.

There are four types of polygons:

1. Concave Polygon
2. Convex Polygon
3. Regular Polygon
4. Irregular Polygon

1. Concave Polygon: A polygon in which at least one interior angle is more than \({180^ \circ }\) is called a concave polygon.
2. Convex Polygon: A polygon in which each interior angle is less than \({180^ \circ }\) is called a convex polygon.
3. Regular Polygon: A polygon is said to be regular if it is equiangular and equilateral
4. Irregular Polygon: A polygon whose sides are not equiangular and equilateral is called as irregular polygon.

Interior and Exterior Angles of Some Regular Polygons

Regular Pentagon: A polygon that has five sides is known as a pentagon. A regular pentagon is one in which all sides and angles of a pentagon are equal.

Interior angle of a regular pentagon \( = \frac{{\left({5 – 2} \right) \times {{180}^ \circ }}}{5} = {108^ \circ }\)
The exterior angle of a regular pentagon \( = \frac{{{{360}^ \circ }}}{5} = {72^ \circ }\)

Regular Hexagon: A polygon that has six sides is known as a hexagon. A regular hexagon is one in which all sides and angles of a hexagon are equal.

Interior angle of a regular hexagon \( = \frac{{\left({6 – 2} \right) \times {{180}^ \circ }}}{6} = {120^ \circ }\)
The exterior angle of a regular hexagon \( = \frac{{{{360}^ \circ }}}{6} = {60^ \circ }\)

Regular Heptagon: A polygon that has seven sides is known as a heptagon. A regular heptagon is one in which all sides and angles of a heptagon are equal.

Interior angle of a regular heptagon \( = \frac{{\left({7 – 2} \right) \times {{180}^ \circ }}}{7} = {\left({\frac{{900}}{7}} \right)^ \circ } \cong {128.57^ \circ }\)
The exterior angle of a regular heptagon \( = \left({\frac{{{{360}^ \circ }}}{7}} \right) \cong {51.43^ \circ }\)

Regular Octagon: A polygon that has eight sides is known as an octagon. A regular octagon is one in which all sides and angles of an octagon are equal.

Interior angle of a regular octagon \( = \frac{{\left({8 – 2} \right) \times {{180}^ \circ }}}{8} = {135^ \circ }\)
The exterior angle of a regular octagon \( = \frac{{{{360}^ \circ }}}{8} = {45^ \circ }\)

Regular Nonagon: A polygon that has nine sides is known as a nonagon. A regular nonagon is one in which all sides and angles of a nonagon are equal.

Interior angle of a nonagon \( = \frac{{\left({9 – 2} \right) \times {{180}^ \circ }}}{9} = {140^ \circ }\)
The exterior angle of a nonagon \( = \frac{{{{360}^ \circ }}}{9} = {40^ \circ }\)

Regular Decagon: A polygon that has ten sides is known as a decagon. A regular decagon is one in which all sides and angles of a decagon are equal.

Interior angle of a decagon \( = \frac{{\left({10 – 2} \right) \times {{180}^ \circ }}}{10} = {144^ \circ }\)
The exterior angle of a decagon \( = \frac{{{{360}^ \circ }}}{10} = {36^ \circ }\)

Note: We can observe an interesting fact that from an equilateral triangle to a regular decagon, the interior angle measure is increasing, and the measure of the exterior angle is decreasing.

Perimeter and Area of a Regular Polygon

The measure 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 measure of its sides.

If \(s\) denotes the side of a regular polygon, then the perimeter of each regular polygon will be as follows.
The perimeter of an equilateral triangle \( = 3s\)
The perimeter of a square \( = 4s\)
The perimeter of a regular pentagon \( = 5s\)
The perimeter of a regular hexagon \( = 6s\)
The perimeter of a regular heptagon \( = 7s\)
The perimeter of a regular octagon \( = 8s\)
The perimeter of a regular nonagon \( = 9s\)
The perimeter of a regular decagon \( = 10s\)
The area of a polygon is the amount of space occupied by it.
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.

Solved Examples – Regular Polygon

Q.1. Is it possible to have a regular polygon, each of whose exterior angles is
a. \( = {25^ \circ }\)
b. \( = {35^ \circ }\)
c. \( = {45^ \circ }\)
Ans: a. Given exterior angle \( = {25^ \circ }\)
We know that the exterior angle of a polygon of \(n\) number of sides is given by \(\frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow {25^ \circ } = \frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow n = \frac{{{{360}^ \circ }}}{{{{25}^ \circ }}}\)
\( \Rightarrow n = 14\frac{2}{5}\) which is not a whole number.
So, it is impossible to have a regular polygon, each of whose exterior angles is \({25^ \circ }.\)
b. Given exterior angle \( = {35^ \circ }\)
We know that the exterior angle of a polygon of \(n\) number of sides is given by \(\frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow {35^ \circ } = \frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow n = \frac{{{{360}^ \circ }}}{{{{35}^ \circ }}}\)
\( \Rightarrow n = 10\frac{{10}}{{35}}\) which is not a whole number.
So, it is impossible to have a regular polygon, each of whose exterior angles is \({35^ \circ }.\)
c. Given: Exterior angle \( = {45^ \circ }\)
We know that the exterior angle of a polygon of \(n\) number of sides is given by \(\frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow {45^ \circ } = \frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow n = \frac{{{{360}^ \circ }}}{{{{45}^ \circ }}}\)
\( \Rightarrow n = 8\)
So, it is possible to have a regular polygon, each of whose exterior angles is \({45^ \circ },\) and it is a regular octagon.

Q.2. Is it possible to have a regular polygon, each of whose interior angles is
a. \({45^ \circ }\)
b. \({60^ \circ }\)
c. \({75^ \circ }\)
Ans: a. Given interior angle \( = {45^ \circ }\)
We know that the interior angle of a polygon of \(n\) number of sides is given by \(\frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}.\)
\( \Rightarrow {45^ \circ } = \frac{{\left({n – 2} \right){{180}^ \circ }}}{n}\)
\( \Rightarrow n \times {45^ \circ } = n \times {180^ \circ } – {360^ \circ }\)
\( \Rightarrow 360 = 180\,n – 45\,n\)
\( \Rightarrow 360 = 135\,n\)
\( \Rightarrow n = \frac{{360}}{{135}}\)
\( \Rightarrow n = \frac{8}{3} = 2\frac{2}{3}\) which is not a whole number.
So, it is impossible to have a regular polygon, each of whose interior angles is \({45^ \circ }.\)
b. Given interior angle \( = {60^ \circ }\)
We know that the interior angle of a polygon of \(n\) number of sides is given by \(\frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}.\)
\( \Rightarrow {60^ \circ } = \frac{{\left({n – 2} \right){{180}^ \circ }}}{n}\)
\( \Rightarrow n \times {60^ \circ } = n \times {180^ \circ } – {360^ \circ }\)
\( \Rightarrow 360 = 180\,n – 60\,n\)
\( \Rightarrow 360 = 120\,n\)
\( \Rightarrow n = \frac{{360}}{{120}}\)
\( \Rightarrow n = 3\)
So, it is possible to have a regular polygon, each of whose interior angles is \({60^ \circ },\) and it is an equilateral triangle.
c. Given interior angle \( = {75^ \circ }\)
We know that the interior angle of a polygon of \(n\) number of sides is \(\frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}.\)
\( \Rightarrow {75^ \circ } = \frac{{\left({n – 2} \right){{180}^ \circ }}}{n}\)
\( \Rightarrow n \times {75^ \circ } = n \times {180^ \circ } – {360^ \circ }\)
\( \Rightarrow 360 = 180\,n – 75\,n\)
\( \Rightarrow 360 = 105\,n\)
\( \Rightarrow n = \frac{{360}}{{105}}\)
\( \Rightarrow n = \frac{{24}}{7}\)
\( \Rightarrow n = 3\frac{3}{7}\)which is not a whole number.
So, it is impossible to have a regular polygon, each of whose interior angles is \({75^ \circ }.\)

Q.3. Find the measure of each interior angle of a regular polygon of \(12\) sides.
Ans: We know that the interior angle of a polygon of n number of sides is \(\frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}\)
Interior angle of a \(12\) sided regular polygon \( = \frac{{\left({12 – 2} \right) \times {{180}^ \circ }}}{{12}} = {150^ \circ }\)
Therefore, each interior angle of a \(12\) sided regular polygon is \({150^ \circ }.\)

Q.4. What is the minimum interior angle possible for a regular polygon?
Ans: As the number of sides of a regular polygon decrease, each of its exterior angles increases and therefore, each interior angle decreases.
Thus, the smallest regular polygon, an equilateral triangle, will have a minimum interior angle, which is \({60^ \circ }.\)

Q.5. What is the maximum exterior angle possible for a regular polygon?
Ans: As the number of sides of a regular polygon decrease, each of its exterior angles increases. So, it is maximum in the case of an equilateral triangle, which is \({120^ \circ }.\)

Summary

This article discussed the definition of polygons, types of polygons, regular polygon and the formula to find the interior and exterior angles of a regular polygon and solved examples related to the regular polygon.

Frequently Asked Questions (FAQs) – Regular Polygon

Let’s look at some of the frequently asked questions about Regular Polygons:

Q.1. Is pentagon a regular polygon?
Ans: A regular polygon is both equiangular and equilateral. So, a pentagon can be a regular polygon if it has equal interior angles and equal sides.

Q.2. What is the area of a regular polygon?
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. How many angles are there in a regular polygon, and how to find their measure in degrees?
Ans: Each regular polygon will have an interior angle and an exterior angle adjacent to it. We have two separate formulas to find the degree measure of an interior angle and exterior angle. If there are n sides in a polygon, then
Interior angle of a regular polygon \( = \frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}\)
The exterior angle of a regular polygon \( = \frac{{{{360}^ \circ }}}{n}\)

Q.4. What is the exterior angle of a \(15\) sided regular polygon?
Ans: We know that exterior angle \( = \frac{{{{360}^ \circ }}}{n}\)
\( \Rightarrow \)exterior angle \( = \frac{{{{360}^ \circ }}}{{15}}\)
\( \Rightarrow \)exterior angle \( = {24^ \circ }\)
Therefore, the exterior angle of a \(15\) sided polygon is \( = {24^ \circ .}\)

Q.5. How do you find the sides of a regular polygon?
Ans: We can find the sides of a regular polygon if we have the measure of an exterior angle. If \(n\) is the number of sides, then
\(n = \frac{{{{360}^ \circ }}}{{exterior\,angle}}\)

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