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November 11, 2024**Properties of a Polygon: **Polygons are described as two-dimensional closed figures created by joining three or more than three line segments. In other words, a closed figure formed of only line segments is called a polygon.

We get to know about polygons mostly while we study geometry. The polygons can be regular, irregular, convex, and concave. We can classify them based on their number of sides and their angles. Polygons have some set of rules or properties. In this article, we will know about polygons and their identification and properties using charts in detail.

In geometry, the definition of a polygon** **is given by a simple closed curve made by only straight-line segments.

The below figures are not polygons.

We can break the word “polygon” into two parts, such as “poly” and “gon”. “Poly” means many, and “gon” means angles. “Poly” and “gon” are Greek words. Every day we deal with different polygons such as triangles, quadrilaterals, pentagons, hexagons, etc.

Each polygon has a unique name based on its number of sides. For instance, the pentagon, “Penta”, means five, and “gon” means angles. So, the pentagon is a polygon with five angles and five sides. This table shows the name of polygons based on the number of their sides.

We can identify or classify a polygon based on the length of its sides to regular and irregular polygons.

A polygon that is equilateral and equiangular is called a regular polygon. In contrast, a polygon whose sides are not equilateral and equiangular is mentioned as an irregular polygon.

In other words, a regular polygon has all identical angles at each vertex, and all sides have the same length. In contrast, a polygon that does not have identical sides and different angles at each vertex is described as an irregular polygon.

A polygon in which at least one interior angle is more than \(180^{\circ}\) is called a concave polygon, while in a convex polygon, each interior angle is less than \(180^{\circ}\).

Two types of angles exist in a polygon. These are exterior angles and interior angles.

**Interior angles** are those angles formed inside the polygon at the vertices.

A polygon’s interior angle is the angle formed between two adjacent sides of the polygon. The interior angle of a polygon is the angle measured at the inside part of the polygon.

**Exterior angles **are the angles formed outside of a polygon when one of its sides is extended. It is adjacent to (beside) the interior angle.

Let us have a look at the below image and understand.

In a regular polygon, all angles are equal and less than \(180^{\circ}\). So, the regular polygons are always convex polygons.

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

(i) Sum of all exterior angles \(=360^{\circ}\)

(ii) Each exterior angle \(=\frac{360^{\circ}}{n}\)

(iii) Sum of all interior angles \(=(n-2) \times 180^{\circ}\)

(iv) Each interior angle \(=\frac{(n-2) \times 180^{\circ}}{n}\)

For an equilateral triangle, \(n=3\)

The measure of each exterior angle of an equilateral triangle \(=\frac{360^{\circ}}{3}=120^{\circ}\)

The measure of each interior angle of an equilateral triangle \(=\frac{(3-2) \times 180^{\circ}}{3}=60^{\circ}\)

For a regular quadrilateral, that is square, \(n=4\)

The measure of each exterior angle of a square \(=\frac{360^{\circ}}{4}=90^{\circ}\) The measure of each interior angle of a square \(=\frac{(4-2) \times 180^{\circ}}{4}=90^{\circ}\)

For a regular pentagon, \(n=5\)

The measure of each exterior angle of a regular pentagon \(=\frac{360^{\circ}}{5}=72^{\circ}\)

The measure of each interior angle of a regular pentagon \(=\frac{(5-2) \times 180^{\circ}}{5}=108^{\circ}\) Similarly, we can find these angle measurements for other regular polygons also.

**Note:** If we apply this formula on regular polygons from an equilateral triangle to a regular decagon, the interior angle increases, and the measure of the exterior angle decreases. Irregular polygons have different measures of interior and exterior angles.

A line connecting a vertex to a non-adjacent vertex of a polynomial is referred to as its diagonal. A triangle doesn’t have any non-adjacent sides, and hence a triangle has zero diagonals. It is impossible to draw a line from one interior angle to any other interior angle of the triangle other than its sides.

As a quadrilateral has four sides, it has two pairs of non-adjacent sides. Hence, a quadrilateral has two diagonals. We can draw the diagonals of a quadrilateral by joining opposite vertices by the line segments. In the same way, we can find five diagonals in a pentagon and so on.

Now, in a polygon, the number of diagonals is increased if the number of sides is increased. It is difficult to draw and count the number of diagonals for a polygon with six sides or more than that.

Luckily, there is a simple formula for calculating how many diagonals a polygon has. Each vertex or corner of a polygon is connected to two adjacent vertices by its sides. These line segments cannot be considered diagonals.

The formula for finding the diagonal of the polynomial with \(n\) sides \(=\frac{n(n-3)}{2}\).

For instance, the quadrilateral only has two diagonals.

For a hexagon, the number of sides is \(n=6\)

Thus, the number of diagonals can be determined by using the above formula.

Therefore, the number of diagonals in a pentagon \(=\frac{6(6-3)}{2}=\frac{6 \times 3}{2}=9\)

** Q.1. Find the measure of each interior angle of a regular polygon of** \(15\)

Interior angle of a \(15\) sided regular polygon \(=\frac{(15-2) \times 180^{\circ}}{15}=156^{\circ}\)

Therefore, each interior angle of a \(15\) sided regular polygon is \(156^{\circ}\).

** Q.2. How many sides does a regular polygon have if the sum of the interior angles is** \(540^{\circ}\)

The formula to measure each interior angle of a regular \(n\)-sides polygon \(=\frac{(n-2) 180^{\circ}}{n}\).

\(\Rightarrow 540^{\circ}=(n-2) 180^{\circ}\)

\(\Rightarrow(n-2)=\frac{540^{\circ}}{180^{\circ}}=3\)

\(\Rightarrow(n-2)=3\)

\(\Rightarrow n=3+2\)

\(\Rightarrow n=5\)

Therefore, the polygon has \(5\) sides.

*Q.3. Find the measure of each exterior angle of a regular octagon.*** Ans:** We need to find the measure of each exterior angle of a regular octagon.

We know that the number of sides of an octagon is, \(n=8\).

The measure of each exterior angle of a \(n\)-sided polygon \(=\frac{360^{\circ}}{8}\)

Therefore, each exterior angle of a regular octagon \(=\frac{360^{\circ}}{8}=45^{\circ}\)

** Q.4. How many sides does a polygon have if it has** \(90\)

The number of diagonals \(=90\).

We know that, number of diagonals in a polygon \(=\frac{n(n-3)}{2}\)

Where \(n\) is the number of sides of the polygon

\(\frac{n(n-3)}{2}=90\)

\(n(n-3)=180\)

\(n^{2}-3 n-180=0\)

\((n-15)(n+12)=0\)

\(n=15 ; n=-12\)

Since sides cannot be negative, the value of \(n\) is \(15\).

** Q.5. A polygon has** \(27\)

Where \(n\) is the number of sides of the polygon.

Now, according to the given question,

\(\frac{n(n-3)}{2}=27 n(n-3)=54 n^{2}-3 n-54\)

\(=0 n^{2}-9 n+6 n-54=0 n(n-9)-6(n-9)=0(n+6)(n-9)=0 n=9,-6\)

Since \(n\) is a negative value, the number of sides present in the polygon is \(9\).

In this article, we studied the definition of polynomials, types of polynomials, angle property, and diagonal property of polynomials. We learnt that we name the polynomials based on their number of sides. We solved some examples of properties of polynomials, such as angles and diagonal properties.

*Q.1. What are the angle properties of polygons?*** Ans:** Formula of each exterior angle of a polygon \(=\frac{360^{\circ}}{n}\) when, the polygon has \(n\) number of sides.

The formula of each interior angle \(=\frac{(n-2) \times 180^{\circ}}{n}\) when, the polygon has \(n\) number of sides.

*Q.2. What is the polygon diagonal formula?*** Ans:** The polygon formula for finding the number of diagonals is \(\frac{n(n-3)}{2}\) where \(n\) is the number of sides of the polygon.

*Q.3. Give some examples of regular polygons?*** Ans:** A polygon having all identical sides and all equal angles are called a regular polygon. Types of regular polygons are:

1. Equilateral triangle

2. Square

3. Pentagon with equal sides

*Q.4. Why is the sum of exterior angles of a polygon 360?*** Ans:** A polygon’s exterior angles and interior angles make a linear pair, and hence they are supplementary. The interior angles add up to \(180(n-2)^{\circ}\), and the sum of the exterior angles is supplementary to this interior angle sum. So, the sum of any polygon’s exterior equals \(360^{\circ}\). Remember here that we are just talking about convex polygons here.

** Q.5. What are regular and irregular polygons?**A polygon that is equilateral and equiangular is called a regular polygon. In contrast, a polygon whose sides are not equilateral and equiangular is mentioned as an irregular polygon.

Ans:

In other words, a regular polygon has all identical angles at each vertex, and all sides have the same length. In contrast, a polygon that does not have identical sides and measurement of angles at each vertex different is described as an irregular polygon.

*We hope this detailed article on the properties of a polygon helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!*