What is a Pentagon?
Geometry is a branch of Mathematics which deals with the study of shapes and their properties. A polygon is 2-dimensional figure which has straight lines connected together to form a closed shape. It is important to note that a polygon does not have curved sides. The straight lines form the sides and angles of the polygon. The word ‘poly-’ means many and ‘-gon’ means angle. Based on the number of sides a polygon has, they are named differently. For instance, a triangle is a polygon with 3 sides, a rectangle is a polygon with 4-sides.
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).
Types of Pentagons:
Regular and Irregular Pentagons:
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.
Convex and Concave Pentagons:
Pentagons can also be classified into 2 types based on their angle measurements. The 2 types are Concave Pentagon and Convex Pentagon.
But if all the interior angles of a pentagon are lesser than 180°, then such a pentagon is known as the Convex Pentagon as shown in the figure below.
· 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.
If one or more of the interior angles of a pentagon has a measure greater than 180°, then such a pentagon is known as the Concave pentagon. These are opposite to the convex pentagons.
· 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.
Angles of a Pentagon:
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.
If a convex regular polygon has ‘n’ sides, then the sum of all its interior angles, S = (n – 2) * 180°
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° = 540°
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 = (n – 2)/ n * 180°
Each Interior angle of Regular Convex Pentagon = (5 – 2) / 5 * 180° = 108°
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 = 360°/n
Each Exterior angle of Regular Convex Pentagon = 360°/5 = 72°
4) Diagonals of a polygon: Diagonal of a polygon is a line segment that joins any two non-adjacent vertices.
Number of diagonals in a polygon of ‘n’ sides = n * (n – 3)/ 2
Therefore, number of diagonals in a pentagon of 5 sides = 5 * (5 – 3)/2 = 5 diagonals.
Perimeter of a Pentagon:
Perimeter of a polygon (regular or irregular) can be easily calculated by simply adding up all the side lengths of the polygon.
Perimeter of a Pentagon = Sum of all the side lengths of the pentagon
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 = 5s
Example: Calculate the perimeter of a regular pentagon whose side length is 6m.
Given side length, s = 6m
Perimeter, P = 5 * s = 5 * 6m = 30m
Area of a Pentagon:
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:
Area of a Regular Polygon, A = 1/2 * Apothem * Perimeter
(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:
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°)
Area of a Regular Polygon, A = 1/2 * Apothem * Perimeter
Therefore, Area of a Regular Pentagon, A
= 1/2 * s/2 * tan(54°) * 5s = 5s2/4 * tan(54°)
This can be simplified to Area of a Regular Pentagon, A = 1.72 s2
Area of an irregular polygon:
This can be calculated by dividing the polygon into set of triangles, and then adding the area of each triangle to get the total area of the irregular pentagon.
Example: Calculate the area of a regular pentagon if its side length is 6m.
Area of a Regular pentagon = 1.72 * s2
==> Area = 1.72 * 62
= 1.72 * 36 = 61.92m2
Therefore, area of the given regular pentagon = 61.92m2