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 .

Table of Contents

## 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.

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## 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:

- Each internal angle measures approximately 128.57 degrees.
- Each external angle measures approximately 51.43 degrees.
- Has 7 lines of symmetry and rotational symmetry of order 7.
- 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:

- The internal angles can vary, but their sum is always 900 degrees.
- Does not have lines of symmetry unless specifically constructed to have some symmetrical properties.
- 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:

- Each internal angle is less than 180 degrees.
- Vertices of the heptagon point outwards, and it does not self-intersect.
- 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:

- At least one internal angle is greater than 180 degrees.
- It can have indentations, and some vertices point inwards.
- 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

- First, find the side length:

## FAQs

**What is the sum of the interior angles of a regular heptagon?**

The sum of the interior angles of any heptagon, whether regular or irregular, is 900 degrees.

**How do I find the measure of each interior angle in a regular heptagon?**

Since a regular heptagon has seven equal interior angles, you can divide the total sum of interior angles (900 degrees) by the number of angles (7).Measure of each interior angle = 900 degrees / 7 ≈ 128.57 degrees.

**How do I find the area of a regular heptagon?**

While there’s a more complex formula using trigonometry, a commonly used approximation for the area (A) of a regular heptagon with side length (s) is:A ≈ 3.634 x s

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