A heptagon is a seven-sided polygon, a two-dimensional closed figure with seven sides and seven angles. It is distinguished by its unique structure, which includes different parts that affect its geometry.
Key Parts of Heptagon
The important parts which make a polygon a "heptagon" are as follows:
Heptagon Sides: The line segment that encloses the figure and forms the boundary line of shape is called a side of the heptagon. There are 7 sides in a heptagon shape. These sides can be of any length, to make a heptagonal structure.
Heptagon Angles: A hexagon makes an angle at each of the seven vertices. For a regular heptagon, the sum of interior angles is 900° and the sum of exterior angles is 360°.
Each of the interior angles would be around 128.5 degrees and the exterior angle would be 51.43 degrees.
For an irregular heptagon the interior and exterior angles will vary.
There are two types of angles in a heptagon, which are :
- Interior Angle of Heptagon
- Exterior Angle of Heptagon
- Interior Angles of Heptagon: There are seven interior angles of a Heptagon and the sum of these angles is 900 degrees. In the case of a regular heptagon, each angle is equal to 128.57 degrees.
- Exterior Angles of Heptagon: There are seven exterior angles in a Heptagon. These exterior angles are formed when the sides of the Heptagon are extended outside the Heptagon. The sum of the exterior angle is 360 degrees.
Heptagon Diagonals: As a heptagon has 7 vertices, the total number of diagonals can be 14. Thus, a heptagon has 14 diagonals in total.
Types of Heptagon Shape
Heptagons can be categorized into two groups, which are :
- Based on Side Length
- Based on Angle Measurement
Heptagons Based on Side Length
A heptagon may have the same length for all seven sides and may not have the same length for all seven sides. Hence, based on Side length the Heptagons are classified as follows :
- Regular Heptagon
- Irregular Heptagon
Types of Heptagons based on side length are shown in the figure below:
Regular Heptagon
A heptagon whose sides and angles are equal is known as a regular heptagon. It is the ideal shape which has equal sides and equal angles with no parallel sides.
Regular heptagons have the following properties:
- The sum of the exterior angle is equal to 360 degrees.
- The measure of interior angle is 128 degrees approximately.
- It has 14 diagonals.
- A regular heptagon can be divided into 5 triangles.
- The measure of the central angle is approximately 54 degrees.
- It has a rotational symmetry of order 7.
Irregular Heptagon
Every object with a closed shape and seven sides is a heptagon. These sides can be of different lengths, at any angle. Hence, there cannot be any particular measure for irregular heptagon.
An irregular heptagon doesn't have lines of symmetry.
Heptagon Based On Angle Measurement
Based on the measurement of angles, Heptagon is classified as :
- Concave Heptagon
- Convex Heptagon
Let's learn them in detail.
Concave Heptagon
In a concave heptagon, at least one of the interior angles of all the angles is greater than 180 degrees. Some of the properties are listed below :
- It is a type of irregular heptagon.
- It is asymmetrical in nature.
- It causes one of the corresponding diagonals to fall outside the boundary line of the heptagon.
Convex Heptagon
When all interior angles of a heptagon measure less than 180 degrees it is known as Convex Heptagon. It has the following properties,
- All interior angles measure less than 180 degrees.
- A regular heptagon(equal sides) is always a convex heptagon.
- A convex heptagon can be both a regular and irregular heptagon.
- All diagonals lie inside the heptagon.
- They may or may not be symmetrical in nature.
Heptagon Properties
Key properties of a heptagon are :
Properties of Heptagon |
|---|
| Property | Description |
|---|
| Sides | A heptagon has seven sides, forming a closed structure. |
| Angles | A heptagon has seven angles. The sum of interior angles is 900 degrees (5π radians), and the sum of exterior angles is 360 degrees. |
| Vertex | It has 7 vertices, where each line segment joins. |
| Diagonals | A heptagon has 14 diagonals, which are line segments joining two non-consecutive vertices. |
| TheInterior Angles | The total sum of all angles in a heptagon is 900 degrees, calculated as (n-2) × 180°, where n is the number of sides. |
| Interior Angle Measure | Each angle measure in a regular heptagon is approximately 128.57 degrees, calculated as (n-2) × 180°/n. |
| Exterior Angle Sum | The sum of exterior angles in a heptagon is always 360 degrees, with each exterior angle being approximately 51.43 degrees. |
| Line of Symmetry | A heptagon has rotational symmetry of order 7 and can be symmetrically divided into congruent halves. |
There are two common Formulas of Regular Heptagon, which are :
- Perimeter of Heptagon
- Area of Heptagon
Perimeter of Heptagon
For a regular heptagon, the perimeter of the heptagon can be calculated using the following formula :
Perimeter of Heptagon = 7 × Sides of Heptagon (for Regular Heptagon)
For irregular heptagons, we need to add all seven sides individually.
Area of Heptagon
The area of a heptagon is given by the following formula,
Area of Regular Heptagon = 3.643 × (side)2
How to Draw Heptagon
Constructing a regular heptagon with the perfect 128. 57 degrees of angle and equal length can be a little tricky. Let's follow the steps below, to construct a regular heptagon.
Step 1: Draw a circle, centered at point x. Also, it is better to leave some extra space outside the circle to make the exterior part of heptagon.
Step 2: Draw a radius connecting the center of the circle to its outer boundary. lets name the radius ,AX.
Step 3: Draw another circle centered at A, which intersects the circle at two points, B and C.
Step 4: Connect B and C. The line segment BC bisects AX.
Step 5: Draw third circle centered at C which will intersect at point D.
Step 6: Use your compass to measure the length of CD and create 7 sides.
Step 7: Join the Seven Points to get the final Heptagonal shape. Erase the unnecessary parts.
Step 8: Your Final Heptagon is constructed. There is a probability of negligible error in this method to construct a heptagon.
Symmetry of a Heptagon
- A regular heptagon has 7 lines of reflectional symmetry and 7-fold rotational symmetry.
- 7 lines of reflectional symmetry: Each line passes through a vertex and the midpoint of the opposite side.
A Regular Heptagon with SymmetryRelated Reads:
Solved Examples of the Properties of Heptagon
Let's solve some example problems on the properties of Heptagon.
Example 1: If the area of a regular heptagon is 714 cm2, what is its side?
Solution:
Area of Heptagon = 3.643 × side × side
Here, we have area = 714 cm2
⇒ 714 = 3.643 × side × side
⇒ Side2 = 714/3.63 = 196.69
⇒ Side = √196.69 ≈ 14
When we solve the above equation, we get the side equals to 14 cm, approximately.
Example 2: Find the area of the heptagon whose sid.. is 8cm?
Solution:
We know, Area of heptagon = 3.643 (side)2
⇒ Area of heptagon = 3.643 × 8cm × 8cm = 233.1 cm2
So, the area of heptagon with side 8cm will be 233.1 cm2
Example 3: Find the perimeter of the heptagon whose side is 8cm.
Solution:
Perimeter of heptagon is 7 × side.
Here side = 8 cm
So, Perimeter = 7 × 8 = 56 cm
The perimeter of heptagon with side 8cm will be 56cm.
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