In mathematics, polygons are geometrical shapes having three or more sides. Geometry is the main branch of mathematics in which we study the relationships of surfaces, angles, lines, and solids, we can also study the properties and the measurements of points and lines.
Polygon is very essential for making geometrical shapes in civil engineering. In this article, we will learn all about the polygon and the derivation of the area of it along with a lot of examples.
What is a polygon?
Polygons are stated as the closed curves containing the line segments of two or more lines and none of the line segments are crosses the other line. The simplest polygons are triangles as it contains three-line segments and the quadrilaterals as it contains fou-line segments.
In simple words, polygons are those shapes that come with the joining of straight lines. For example, pentagon, hexagon, heptagon, and octagon. In these examples, these shapes contain five, six, seven, and eight lines respectively.
Polygons not only depend on the straight line for making the shapes although angles among the straight line are also used for making the exact shape of a triangle, quadrilaterals, pentagon, hexagon, or any other shape.
To calculate the area of the polygon, we have a formula.
Area of the polygon = (x2 * n * cot(π/n)) / 4
In this formula, x is the length of the side, n is the total sides present in the shape, and the tangent is an angle among them.
There is another formula for calculating the area of the polygon, with the help of radius and numbers of the sides.
Area of the polygon = n * r2 * tan(π/n)) / 2
How to calculate the area of the polygon?
The following methods are used by engineers to find the area of polygons.
Example 1
Determine the area of the polygon, if the length of the side is 9 and the total sides of the shape are 11.
Solution
Step 1:Take the given information.
Length = x = 9
Total numbers of sides = n = 11
Step 2:Take the general formula to calculate the area of the polygon.
Area of the polygon = (x2 * n * cot(π/n)) / 4
Step 3:Put the given values in the above formula.
Area of the polygon = (92 * 11 * cot(π/11)) / 4
= (9 x 9 * 11 * cot(π/11)) / 4
= (81 * 11 * cot(π/11)) / 4
= (81 * 11 * cot (0.2856)) / 4
= (891 * 3.4057) / 4
= 3034.4787 / 4
Area of the polygon = 758.6197
Example 2
Determine the area of the polygon, if the length of the side is 5 and the total sides of the shape are 6.
Solution
Step 1:Take the given information.
Length = x = 5
Total numbers of sides = n = 6
Step 2:Take the general formula to calculate the area of the polygon.
Area of the polygon = (x2 * n * cot(π/n)) / 4
Step 3:Put the given values in the above formula.
Area of the polygon = (52 * 6 * cot(π/6)) / 4
= (5 x 5 * 6 * cot(π/6)) / 4
= (25 * 6 * cot(π/6)) / 4
= (25 * 6 * cot (0.5236)) / 4
= (150 * 1.7320) / 4
= 259.8 / 4
Area of the polygon = 64.95
To calculate the immediate results of the given problems an online polygon calculator can be used for this. In this calculator you have to simply put the input and your output will be in your front in a few seconds.
Type of the polygon
There are many types of polygon used for different purposes. Let’s discuss some of them.
Regular polygon
In polygons, if all the sides are equal and have equal length and angles among them, then this kind of polygon is said to be the regular polygon. In simple words, a polygon having equal straight lines along with the same interior angles is said to be the regular polygon.
A regular polygon is also written as an equilateral or an equiangular. For example, in the square all the sides are equal having the same length and interior angle of 90o, so we can conclude that the square is a regular polygon.
There isanother regular polygon is present such as hexagon. It is also a regular polygon. Because all the six sides are equal in length and have the same interior angles of 120o among them.
Irregular polygon
In this type of polygon, all the sides and the interior angles among them are unequal. In simple words, a polygon having unequal straight lines along with the different interior angles is said to be the irregular polygon.
For example, an isosceles triangle, a rectangle, and a quadrilateral are irregular polygons. Because in isosceles triangles only two sides are equal while the third one is notthe same so it is an irregular polygon.
While in a rectangle, all the interior angles are the same but the sides are different so we can conclude that the rectangle is an irregular polygon.
Convex polygon
The polygons whose interior angles are less than 180-degree are said to be the convex polygon. In this type of polygon, the interior angles are not greater than 180-degree. In other words, a polygon that is opposite to a concave polygon is said to be the convex polygon.
In convex polygons, the vertices of the shapes are always outwards. For example, a regular hexagon is said to be the convex polygon because all the interior angles are of 120-degree that are less than 180-degree.
Concave polygon
The polygons in which at least one interior angle is more than 180-degree are said to be the concave polygon. In this type of polygon, the interior angles can be smaller than 180-degree but one angle must be more than 180-degree. In other words, a polygon that is opposite to a convex polygon is said to be the concave polygon.
In concave polygons, the vertices of the shapes can be drawn both outwards and inwards.
Summary
The polygons are usually used on a wider scale for construction. These shapes can be regular or irregular. By following the above procedure, you can grab all the basic knowledge about polygons.
