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Now, a polygon which has 5 sides, thus forming 5 angles is known as the Pentagon. The word ‘pentagon’ refers to ‘penta-‘ meaning 5 and ‘-gon’ meaning angle. Pentagons are very commonly observed in various geometric calculations and thus play an important role in geometry. As shown in the diagram below, a pentagon has 5 vertices and 5 edges (or sides).

Pentagons are classified into 2 types based on their side measurements. The 2 types are Regular Pentagon and the Irregular Pentagon. Regular pentagon is a pentagon which has all the 5 sides of equal lengths. This also implies that all the 5 angles of the regular pentagon are equal. However, for an irregular pentagon, all the 5 sides and the 5 angles are not of equal measurement. This can be observed in the diagram as shown below.

Pentagons can also be classified into 2 types based on their angle measurements. The 2 types are Concave Pentagon and Convex Pentagon.

· In a convex pentagon, all the vertices point outward away from the interior of the pentagon.

· A line drawn through a convex pentagon will intersect the pentagon twice.

· All the diagonals of the convex pentagon lie inside the pentagon as shown in the figure below.

· In a concave pentagon, vertex appears to be pushed inside the pentagon.

· A line drawn through a concave pentagon (depending on where the line is drawn) can intersect the pentagon at more than 2 points. The figure below shows that the line drawn intersects the pentagon at 4 points.

· Not all diagonals of a concave pentagon lie inside the pentagon. Some of the diagonals may also lie outside as shown in the figure below.

1) Sum of the Interior Angles of Regular Convex Pentagon:

We can find the sum of all the angles in a regular pentagon as well its each interior and exterior angle.

A pentagon has 5 sides, so applying the above formula we get:

Sum of all the interior angles in a regular convex pentagon, S = (5 – 2) * 180°

2) Interior angles of a Regular Convex Pentagon:

We can find the interior angles of a Regular Convex Polygon of ‘n’ sides by using the formula:

Each Interior angle of Regular Convex Pentagon = (5 – 2) / 5 * 180°

3) Exterior angles of a Regular Convex Pentagon:

We can find the measure exterior angle of a Regular Convex Polygon of ‘n’ sides by using the formula:

Each Exterior angle of Regular Convex Pentagon = 360°/5

4) Diagonals of a polygon: Diagonal of a polygon is a line segment that joins any two non-adjacent vertices.

Therefore, number of diagonals in a pentagon of 5 sides = 5 * (5 – 3)/2 =

Perimeter of a polygon (regular or irregular) can be easily calculated by simply adding up all the side lengths of the polygon.

In case of a regular pentagon, all its sides are equal. If the side length of a regular pentagon = s, then the Perimeter of a Regular Pentagon = s + s + s + s + s =

Given side length, s = 6m

Perimeter, P = 5 * s = 5 * 6m = 30m

Area of a Regular pentagon can be calculated by using different measurement and methods. One of the easiest way to calculate the area of regular pentagon is by using the below formula:

(Note: Apothem of a Polygon is the perpendicular line segment drawn from the center of the polygon to the midpoint of one the polygon’s sides).

Area of a regular pentagon can be calculated by using trigonometry as follows:

Let the side length of the regular pentagon, PQ = s (as shown in the figure)

OM is the Apothem and let its length be = a

MQ = s/2 (as ‘M’ is the midpoint of PQ)

Interior angle of regular pentagon = 108°, hence angle OQM = 108/2°

Therefore, angle OQM = 54°

In triangle OMQ, tan (54°) = Opposite side/ Adjacent side = OM/ MQ

tan(54°) = a/(s/2) ==> tan(54°) = 2a/s ==> a = s/2 * tan(54°)

Therefore,

This can be simplified to Area of a Regular Pentagon,

Area of a Regular pentagon = 1.72 * s

Therefore, area of the given regular pentagon = 61.92m