Area of Pentagon

The Pentagon belongs to the Polygon family of Geometric Shapes. The word “Pentagon” derives its origins from the Greek Word “Pentagonos”. The word “Pentagonos” means five-angled”. The pentagon can be defined as a two-dimensional closed geometric shape with five vertices connected by five sides. Now we will look at the figures of both The regular pentagon and the irregular pentagon to help us understand the difference between them better.

Regular Pentagon vs. Irregular Pentagon-

Now we will see figures pertaining to both The Regular Pentagon and The Irregular Pentagon to help us better understand the topic. 

In a Regular pentagon, all external angles are equal to each other and all internal angles are equal to each other. The measures of the Internal Angles are equal to 108° and the measures of the External Angles are equal to 72°. All the sides of The Regular Pentagon are equal in measure to each other. The sides of The Regular Pentagon are all straight. The sum of the Interior Angles in The Regular Pentagon is equal to 540°.

Now we will look at an Irregular Pentagon.

The above figure depicts an Irregular Pentagon. This figure shows three Irregular Pentagons. Now we will see some properties of The Irregular Pentagon. In The Irregular pentagon, all the angles may be of different measures and the sides of The Irregular Pentagon may be of different sides.

Now we will see the two very common types of Irregular Pentagons:- The Concave Pentagon and the Convex Pentagon. Now we will look at some of the properties of The Convex Pentagon and the concave Pentagon.

In a Convex Pentagon, all the interior angles are less than 180°. Also, in a Convex Pentagon all the corners are bent out of shape. A particular Convex Pentagon can be either a Regular Pentagon or an Irregular Pentagon.

The Perimeter of a Pentagon-

The Perimeter of a particular Geometric Shape or Figure is the length of its boundary.

Now we will move on to the Physical Dimensions of The Pentagon. We will start here with the Perimeter of The Pentagon.

There are different methods to calculate The Perimeters of The Regular Pentagons and The Irregular Pentagons. 

For the Regular Pentagon, if we assume that the length of a side is a,

Perimeter (P) is given by,

P=5*a

P=5a 

Now for the Irregular Pentagon, if the lengths of the sides are a, b, c, d, e

The Perimeter (P) is given by,

P=a+b+c+d+e

We have considered 5 different side lengths because the sides of an Irregular Pentagon might be all different in length.

Area-

The area of a Particular Geometric Shape is the Area inside the boundary of that particular Geometric Shape.

In this case, The Area of a Pentagon is the area of the space enclosed by The Pentagon.

Now we move on to the most important topic in this chapter, which is how to find out the Area of a Pentagon. Finding the Area of a Pentagon is a much more complicated process than finding out The Perimeter.

So, to find out The Area of a Regular Pentagon, The Regular Pentagon must be divided into five triangles. This step can be done by drawing five lines from the centre of the Pentagon.

So now that we have the triangles, we have to find out the Area of one of those triangles. We can find the area of the triangle by any of the two methods.

A= (1/2)*b*h

Where,

 b: length of the base of the triangle.

 h: length of the height of the triangle.

A=√s(s-a)(s-b)(s-c)

Where,

s= semi-perimeter of the triangle.

s= (a+b+c)/2

a: length of the first side of the triangle.

b: length of the second side of the triangle.

c: length of the third side of the triangle.

So, now that we have calculated the area of the triangle, we will multiply it by five to find out The Total Area of The Regular Pentagon. This is because the regular pentagon has five triangles. 

If we have to find out the Area of an Irregular Pentagon, we can draw an appropriate number of triangles, find out the area of those triangles and add up those areas, we can find out The Total Area of the Irregular Polygon.

Now we will solve some problems related to the topics discussed above.

Examples-

5. Find the perimeter of a Regular Pentagon with a side equal to 5.

Soln.

Given:

a=5

P=5a

P=5*5

P=25

So the Perimeter of The Regular Pentagon is 25.

7. Find the perimeter of a Regular Pentagon with a side equal to 7.

Soln.

Given:

a=7

P=5a

P=5*7

P=35

So the perimeter of The Regular Pentagon is 35.

11. Find the Perimeter of an Irregular Pentagon with sides equal to 3, 5, 7, 9 and 11.

Soln.

The perimeter of an Irregular Pentagon is the sum of the measures of its sides.

P=3+5+7+9+11=35

Conclusion

In this chapter, first, we have defined the pentagon. Then we moved on to the Introduction. In the Introduction topic, we saw The Pentagon in detail. Then we saw the difference between The Regular Pentagon and The Irregular Pentagon using images. Then we saw how to calculate the perimeters of both the regular pentagon and the irregular pentagon. After that, we saw how to calculate the area of The Regular Pentagon and The Irregular Pentagon. Then we solved some problems related to these topics.