Geometry - Heptagon
- What is a Heptagon?
- Practical Examples
- Properties of Heptagon
- Types of Heptagons
- Perimeter of a Heptagon
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is a Heptagon?
A heptagon is a two-dimensional polygon with seven line segments and seven angles. A heptagon can be regular or irregular depending upon the congruency of sides and angles.
The term heptagon is derived from the Greek words “hepta” which means seven and “gonia” which means angles. Therefore it means seven angles, which forms a seven-sided shape. In Latin, a heptagon is called a septagon, where septa means seven and gon means angles which again corresponds to the seven-sided polygon.
Practical Examples
The practical examples of heptagons are numerous which includes coins, sign board, jewelry, stamps and tokens, etc.
Properties of Heptagon
- The sum of interior and exterior angles: The sum of the interior angles of a heptagon is `900°`. The sum of the exterior angles of a heptagon is `360°`.
- Interior angles of a regular heptagon: A regular heptagon has all its angles of equal measure, which is approximately `128.57°`.
- Exterior angles of a regular heptagon: A regular heptagon has all its exterior angles that are also congruent, which have a measure of approximately `51.43°`.
- Symmetry: A regular heptagon is symmetrical when it is rotated by some fixed angle, it appears the same. An irregular heptagon is not symmetrical.
- Diagonals: A heptagon has `14` diagonals which are formed by joining the non-adjacent vertices.
Types of Heptagons
- Regular Heptagon: A regular heptagon is a heptagon that has all its sides of equal measure, and angles of equal measure. It is symmetrical and has equal exterior angles.
- Irregular Heptagon: An irregular heptagon is also called a non-regular heptagon whose sides do not measure the same length and whose angles are also not equal.
- Convex Heptagon: A heptagon is called a convex heptagon if all the interior angles are less than `180°`. Its shape looks outward. A convex heptagon is shown in the image below.
- Concave Heptagon: A heptagon is called a concave heptagon in which at least one of its interior angles is greater than `180°`. It can have inward notches or indentations. A concave heptagon is shown below.
Perimeter of a Heptagon
The perimeter of the heptagon can be calculated by simply adding the side lengths. In the case of an irregular heptagon, the perimeter can be given as follows
Perimeter of an irregular heptagon `="Side"_1+"Side"_2+"Side"_3+"Side"_4+"Side"_5+"Side"_6+"Side"_7`
For a regular heptagon, the side lengths are equal, therefore perimeter can be given as follows,
Perimeter of a regular heptagon `= 7×("Side length")`
If the side length of the regular heptagon is `x`, then perimeter `=7x`.
Solved Examples
Example `1`: A regular heptagon with a side length of `7` cm is drawn. Find its perimeter.
Solution:
Here, side length `= 7` cm
Perimeter of the heptagon `= 7×("Side length")`
`= 7×7`
`= 49` `cm`
Example `2`: If an irregular heptagon is measured with its side lengths `4` cm, `6` cm, `5` cm, `8` cm, `4` cm, `6` cm, and `5` cm. Find its perimeter.
Solution:
`"Side"_1=4` `cm`, `"Side"_2=6` `cm`, `"Side"_3=5` `cm`, `"Side"_4=8` `cm`, `"Side"_5=4` `cm`, `"Side"_6=6` `cm`, `"Side"_7=5` `cm`
Perimeter of the irregular heptagon `= "Side"_1+"Side"_2+"Side"_3+"Side"_4+"Side"_5+"Side"_6+"Side"_7`
`= "Side"_1+"Side"_2+"Side"_3+"Side"_4+"Side"_5+"Side"_6+"Side"_7`
`=4+6+5+8+4+6+5`
`=38` `cm`.
Example `3`: The sum of the interior angles of a regular heptagon is `900°`. What will be the measure of each interior angle?
Solution:
The heptagon can be given as shown in the image below.
The sum of interior angles of the heptagon `=∠A+∠B+∠C+∠D+∠E+∠F+∠G=900°`
Here, Let `∠A=∠B=∠C=∠D=∠E=∠F=∠G=x`
`x+x+x+x+x+x+x=900°`
`7x=900°`
`x=(900°)/7`
`x=128.57°`
Hence, each interior angle will measure `128.57°`.
Example `4`: The sum of the exterior angles of a regular heptagon is `360°`. What is the measure of each exterior angle?
Solution:
The heptagon can be given as shown in the image below.
The sum of exterior angles of the heptagon `=∠1+∠2+∠3+∠4+∠5+∠6+∠7=360°`
Here, Let `∠1=∠2=∠3=∠4=∠5=∠6=∠7=y`
`y+y+y+y+y+y+y=360°`
`7y=360°`
`y=(360°)/7`
`x=51.43°`
Hence, each exterior angle will measure `51.43°`.
Example `5`: A regular heptagon has a perimeter of `63` cm. Find the length of its sides.
Solution:
Here, the perimeter of the heptagon `= 63` cm
`"Perimeter of the heptagon" = 7 × "side length"`
`63 =7×"side length"`
`"side length"=63/7`
`"side length"=9` `cm`.
Practice Problems
Q`1`. What is the perimeter of a regular heptagon with a side length of `8` cm?
- `48` `cm`
- `56` `cm`
- `60` `cm`
- `36` `cm`
Answer: b
Q`2`. An irregular heptagon is measured with its side lengths `7` cm, `8` cm, `5` cm, `4` cm, `6` cm, `4` cm, and `9` cm. Find its perimeter.
- `56` `cm`
- `30` `cm`
- `43` `cm`
- `63` `cm`
Answer: c
Q`3`. The perimeter of a regular heptagon is `140` cm. Find out the length of each side of the heptagon.
- `20` `cm`
- `40` `cm`
- `28` `cm`
- `30` `cm`
Answer: a
Q`4`. If an irregular heptagon has an angle measure of `109°`, `184°`, `154°`, `179°`, `78°`, `64°`, and `132°`. Find the sum of all its interior angles.
- `700°`
- `1000°`
- `900°`
- `700°`
Answer: c
Q`5`. A heptagon has `"_____"` sides.
- `4`
- `6`
- `9`
- `7`
Answer: d
Frequently Asked Questions
Q`1`. What is the difference between the interior and exterior angles of a heptagon?
Answer: An interior angle is an angle formed by two adjacent sides of a heptagon. It lies inside the heptagon. Whereas an exterior angle is an angle formed between a side of the heptagon and an adjacent side extended outward.
Q`2`. Are the irregular heptagons not heptagons?
Answer: The irregular heptagons are also heptagons, but the difference between regular and irregular heptagons is that all the sides and angles in an irregular heptagons are not equal.
Q`3`. Is there any formula for calculating the area of the heptagon?
Answer: Yes and it is calculated by the following formula.
Area of a regular heptagon `=1/2a×P`
where `a =` side length of the regular heptagon
and `P=` perimeter of the heptagon