Heptagon

In this topic, we will see some characteristics of the Heptagon Shape. The heptagon consists of seven sides and seven shapes. Each angle in a Regular Heptagon is 128.57°. Now the difference between a Regular Heptagon and an Irregular Heptagon is that in a Regular Heptagon all sides are equal to each other in length and all the angles are equal (128.57°). Whereas in an Irregular Heptagon, the sides are of different lengths and the angles are of different values. In a Regular Heptagon, the sum of all the angles is 900° (899.99° to be precise).

Area of a Heptagon

Area (A) of the Heptagon like the one given in the figure is given as,

A= (7/4)*(a2)*cot (π/7)

A= 3.634*(a2)

The perimeter of a Heptagon

Perimeter (P) of the Heptagon like the one given in the figure above is given by:-

P=7*(a) =7a

Apothem of a Regular Heptagon

Apothem of a Regular Heptagon is a numerical attribute which is basically the line drawn from the center of the Heptagon to the center of one of the sides of the Heptagon.

It is given by,

Suppose we take L as the symbol for Apothem,

L=(s/2)*(tan (180/n))

Where,

s: length of the side

n: no of sides

In the case of Heptagon, n will be equal to 7

For example, if we had to find out the Apothem of a Heptagon with the side as 7,

L= (7/2)*(tan (180/7))

L=7.268

Properties of a Heptagon

1. An Exterior Angle of the Regular Heptagon is equal to 51.43°.

2. An Interior Angle of the Regular Heptagon is equal to 128.57°.

3. A Heptagon has seven sides.

4. A Heptagon has seven vertices.

5. In a Regular Heptagon, all the sides are of equal length.

6. In an Irregular Heptagon, all the sides are not of equal length.

7. In a Regular Heptagon, all the angles are of equal measure and their value is 128.57°.

8. In an Irregular Heptagon, all the angles are not of equal measure.

9. A Heptagon is also referred to as a Septagon sometimes.

10. However the sum of angles in both Regular Heptagon and Irregular Heptagon will be equal to 900°.

11. A Heptagon has 14 diagonals because it has 7 vertices.

12. The Area of a Regular Heptagon is given by A= 3.634*(a2).

13. The Perimeter of a Regular Heptagon is given by P=7*a.

14. The Apothem of a Regular Heptagon is given by L=(s/2)*(tan (180/n)).

15. Heptagons are also categorized into Concave and Convex Heptagons.

16. Concave Heptagons are Heptagons that have at least one vertex pointed inwards.

17. Convex Heptagons have all the vertices pointing outwards.

18. Concave Heptagons have an interior angle that is more than 180°.

19. No interior angle of a Convex Heptagon is greater than 180°.

20. All the diagonals of the Convex Heptagon lie inside the Heptagon.

21. At least one Diagonal of the Concave Heptagon will lie outside the closed shape.

22. A Convex Heptagon can be both a Regular Heptagon and an Irregular Heptagon.

23. All concave Heptagons are Irregular Heptagons.

Solved Examples:-

Q. The side of a Regular Heptagon is equal to 6 find its Area and Perimeter.

Soln. 

Given in the question a=6

A=3.634*(a2) =3.634*(62)-3.634*36=130.824

P=7*a=7*6=42

So the Area of a Heptagon with side 6 is 130.824 and the Perimeter is 42.

Q. The side of a Regular pentagon is 12; find its Area and Perimeter.

Soln.

Given in the question a=12

A=3.634*(a2) =3.634*(122) =3.634*144=523.296

P=7*a=7*12=84

So the Area of a Heptagon with side 12 is 523.296 and its Perimeter is 84.

Q. The Area of a Regular Heptagon is 363.4, find its side.

Soln. 

A=3.634*(a2) =363.4

363.4=3.634*(a2)

a2= (363.4/3.634)

a2= 100

a= 10

So the side of a Regular Heptagon with an Area equal to 363.4 is 10.

Q. The Perimeter of a Regular Heptagon is 49; find the length of its side.

Soln.

P=7*a=49

7*a=49

a= (49/7)

a=7

So the side of a Regular Heptagon with Perimeter Equal to 49 is 7.

Conclusion

In the chapter on Heptagons first, we started by describing the Heptagon briefly. We also saw how the term was coined.

The Introduction Topic was a huge one. It consisted of many sub-topics. We first saw how to calculate the Area of a Heptagon. Then we learned how to calculate the Perimeter of the Heptagon. We also saw a small sub-topic called The Apothem of a Polygon. The Apothem is nothing but the line drawn from the center of the Heptagon to the midpoint of one of its sides.

Then we saw the many properties of a Heptagon. In these properties, we saw the physical dimensions of a Heptagon, categorization of Heptagons, etc. among many other things.

Then we solved a few problems pertaining to every sub-topic in the Introduction Topic.