Regular Heptagon: Definition, Formulas, and Solved Examples

A regular heptagon is a seven-sided polygon where all sides and angles are equal. Understanding a regular heptagon involves knowing its properties, such as internal and external angles, and the formulas to calculate its area and perimeter. The internal angle of a regular heptagon can be determined using the formula (n−2)×180 degrees/n, where n is the number of sides, which gives us an internal angle of approximately 128.57 degrees for a heptagon. The external angle is 360 degrees divided by 7, approximately 51.43 degrees. These concepts and their applications are commonly tested in competitive exams such as the JEE, NEET, SSC, CAT, MAT, PSAT, GRE, Olympiads, and various engineering entrance exams. By exploring solved examples, you gain a better understanding of how to work with these geometric figures .

Definition of Regular Heptagon

A regular heptagon is a polygon with seven equal sides and seven equal interior angles. It is a two-dimensional shape that is closed and convex. Here are some important properties of a regular heptagon:

• Number of sides: 7
• Number of angles: 7
• The sum of interior angles: 900 degrees
• Measure of each interior angle: Approximately 128.57 degrees
• The sum of exterior angles: 360 degrees (like any polygon)  
• Measure of each exterior angle: Approximately 51.43 degrees  
• Number of diagonals: 14
• Symmetry: It has rotational symmetry of order 7.

Also Read: Find the Area of the Shaded Region: Square, Rectangle, Circle and Triangle 

Types of Heptagon

Heptagons, being seven-sided polygons, can be categorized based on the equality of their sides and angles. Here are the main types of heptagons:

Regular Heptagon

Definition: A heptagon where all seven sides are of equal length, and all internal angles are equal.

Properties:

1. Each internal angle measures approximately 128.57 degrees.
2. Each external angle measures approximately 51.43 degrees.
3. Has 7 lines of symmetry and rotational symmetry of order 7.
4. The area can be calculated using the formula: 

Area = 7/4a²cot⁡(π/7)

Where a is the side length.

Irregular Heptagon

Definition: A heptagon where the sides and/or angles are not all equal.

Properties:

1. The internal angles can vary, but their sum is always 900 degrees.
2. Does not have lines of symmetry unless specifically constructed to have some symmetrical properties.
3. The area can be found using various methods such as the triangulation method or using the coordinates of vertices if available.

Convex Heptagon

Definition: A heptagon where all internal angles are less than 180 degrees.

Properties:

1. Each internal angle is less than 180 degrees.
2. Vertices of the heptagon point outwards, and it does not self-intersect.
3. The sum of the internal angles is 900 degrees.

Concave Heptagon

Definition: A heptagon where one or more internal angles are greater than 180 degrees.

Properties:

1. At least one internal angle is greater than 180 degrees.
2. It can have indentations, and some vertices point inwards.
3. The sum of the internal angles is still 900 degrees, but the shape appears more complex due to the concavity.

Properties of Regular Heptagon with Formulas

A regular heptagon is a seven-sided polygon with equal sides and equal internal angles. Here are the key properties and formulas associated with a regular heptagon.

1. Sides and Angles:

• Number of Sides (n): 7
• Length of Each Side (a): All sides are equal in length.
• Internal Angle: 

Internal Angle = (n−2)×180∘/ n = 5×180∘/7 ≈ 128.57∘

• External Angle: 

External Angle = 360∘/ n = 360∘/7 ≈ 51.43∘

2. Perimeter:

• The perimeter (P) of a regular heptagon is the sum of the lengths of all its sides. 

Perimeter = 7×a

3. Area:

• The area (A) of a regular heptagon can be calculated using the following formula: 

Area = 7/4a²cot⁡(π/7)

where a is the length of a side, and cot⁡\cotcot is the cotangent function.

4. Circumradius:

• The radius (R) of the circumcircle (the circle that passes through all the vertices) can be found using: 

R = a / 2sin⁡(π/7)

5. Symmetry:

• A regular heptagon has 7 lines of symmetry.
• It has rotational symmetry of order 7, meaning it looks the same after a rotation of 360∘/7 ≈ 51.43∘.

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Properties of Regular Heptagon With Solved Examples

Here, we will explore the properties of Regular Heptagon formulas and provide five solved examples.

Example 1: Perimeter

• Problem: Find the perimeter of a regular heptagon with a side length of 8 cm.
• Solution:
  • P = 7 x s
  • P = 7 x 8 cm
  • P = 56 cm

Example 2: Area

• Problem: Calculate the area of a regular heptagon with a side length of 6 inches.
• Solution:
  • A ≈ 3.634 x s²
  • A ≈ 3.634 x (6 inches)²
  • A ≈ 130.824 square inches

Example 3: Interior and Exterior Angles

• Problem: What is the measure of each interior and exterior angle of a regular heptagon?
• Solution:
  • Interior angle ≈ 128.57 degrees
  • Exterior angle ≈ 51.43 degrees

Example 4: Finding Side Length from Perimeter

• Problem: The perimeter of a regular heptagon is 91 cm. Find the length of each side.
• Solution:
  • P = 7 x s
  • 91 cm = 7 x s
  • s = 91 cm / 7
  • s = 13 cm

Example 5: Finding Area from Perimeter

• Problem: The perimeter of a regular heptagon is 42 cm. Find its area.
• Solution:
  • First, find the side length:
    • P = 7 x s
    • 42 cm = 7 x s
    • s = 6 cm
  • Then, find the area:
    • A ≈ 3.634 x s²
    • A ≈ 3.634 x (6 cm)²
    • A ≈ 130.824 square cm

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