What is a Pentagon

What is a Pentagon 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. 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. Properties: 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. Concave Pentagon: 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. Properties: 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 polygons 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 (approximately) 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