Have you ever wondered about heptagons and their fascinating properties? In this article, we'll explore what makes a heptagon unique, its different types, and its real-life examples. So, let's dive in!
What is a Heptagon?
A heptagon is a two-dimensional shape with 7 sides and 7 angles. It belongs to the class of polygons in two-dimensional geometry. Polygons are closed shapes made up of straight lines and no curves.
The word "heptagon" is formed by combining "hepta" (meaning seven) and "gonia" (meaning angle) in Greek. In Latin, heptagon is known as a septagon, where "septa" means seven and "gon" means angle.
Properties of Heptagon
Let's explore some interesting properties of heptagons:
- It has seven sides, seven angles, and seven vertices.
- The sum of the interior angles of a heptagon is 900°.
- It has 14 diagonals.
Types of Heptagon
There are two main types of heptagons based on their shapes:
- Regular Heptagon: It has equal sides and equal angles. All the angles are 128.57°, and all the sides have the same length. There are no parallel sides.
Properties of a Regular Heptagon:
- The sum of its exterior angles is 360°.
- The measure of each interior angle is approximately 128.57°.
- The central angle of a regular heptagon measures about 51.43°.
- A regular heptagon has 14 diagonals.
- Irregular Heptagon: It has sides and angles that vary in length and degree, respectively. It can have two or more pairs of parallel sides.
For example, take a look at the coin and the arrow in the image below:
Both objects have 7 sides and are closed, making them heptagonal in shape. However, the coin is a regular heptagon, with all sides and angles equal, while the arrow is an irregular heptagon with sides of different lengths and angles.
Concave and Convex Heptagon
There are two other types of heptagons to explore:
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Concave heptagon: At least one of its angles points inward, while the other angles point outward. It can have one or more interior angles greater than 180°.
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Convex heptagon: All the interior angles are less than 180°, and they seem to be pointing outward. A regular heptagon is always convex.
Perimeter of a Heptagon
The perimeter of a heptagon refers to the total length of its boundary. For a polygon, the perimeter is given by the sum of all its sides.
Let's say we have a regular heptagon with each side measuring "a" units. The perimeter will be given by the formula:
Perimeter = 7 times a units.
Real-life Examples
In real life, heptagons can be found in various objects. Here are a couple of examples:
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A cactus: Certain cacti, when viewed from the top, form heptagons.
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A greenhouse: The front of a greenhouse often has seven sides and seven angles, making it a heptagon.
Solved Examples
Let's solve some example problems to solidify our understanding:
1. Find out the perimeter of a regular heptagon with a side of 15 cm.
Solution: The perimeter of a regular heptagon with side length "a" is given by: Perimeter = 7 times a.
For the given heptagon, a = 15 cm. Therefore, perimeter = 7 times 15 = 105 cm.
2. Find the perimeter of an irregular heptagon with sides measuring 7 cm, 8 cm, 9 cm, 10 cm, 11 cm, 12 cm, and 13 cm.
Solution: Perimeter is given by the sum of all sides. Therefore, the perimeter of the given irregular heptagon = 7 cm + 8 cm + 9 cm + 10 cm + 11 cm + 12 cm + 13 cm = 70 cm.
3. What is the side of a regular heptagon with a perimeter of 224 cm?
Solution: The perimeter of a regular heptagon with side length "a" is given by: Perimeter = 7 times a. It is given that the perimeter = 224 cm. Therefore, Perimeter = 7 times a = 224 cm. So, a = 224/7. a = 32 cm.
Practice Problems
Now that you have a good understanding of heptagons, why not test yourself with some practice problems? Try solving problems related to heptagons to strengthen your knowledge!
Frequently Asked Questions
If you have any questions or doubts about heptagons, here are some commonly asked questions that might provide the answers you're looking for.