A regular pentagon is a 5-sided polygon with all sides of equal length and all interior angles equal, each measuring 108 degrees. It is a symmetric and closed geometric shape often found in nature and design. The properties of a regular pentagon include equal side lengths, equal angles, and rotational symmetry of order 5, meaning it looks the same after a rotation of 72 degrees. Additionally, the diagonals of a regular pentagon intersect each other at equal angles and create smaller pentagons and triangles within. Understanding a regular pentagon involves knowing specific formulas, such as those for calculating the area and perimeter. Through solved examples, you can gain a clear understanding of the geometric characteristics of pentagons and excel in **competitive exams** such as **JEE**, **NEET**, **SSC**, **CAT**, **MAT**, and various **engineering entrance tests**. It also helps to reinforce learning and application in various fields such as mathematics, architecture, and art.

Table of Contents

## Definition of Regular Pentagon

A regular pentagon is a five-sided polygon where all sides are of equal length and all interior angles are equal, each measuring 108 degrees. Here are some important properties of a regular pentagon:

**Equal Sides and Angles**: All five sides are of equal length, and all five interior angles are equal, each being 108 degrees.**Symmetry**: A regular pentagon has rotational symmetry of order 5, meaning it looks the same after a rotation of 72 degrees (360 degrees divided by 5). It also has five lines of symmetry, each passing through a vertex and the midpoint of the opposite side.**Diagonals**: A regular pentagon has five diagonals. Each diagonal intersects at the golden ratio, approximately 1.618, dividing each other into segments that are in the golden ratio.**Circumcircle**: A regular pentagon can be inscribed in a circle (circumcircle), with all its vertices lying on the circle.**Area and Perimeter**: The area A of a regular pentagon with side length s can be calculated using the formula**Area = ¼ √5(5+2√5)s²**. The perimeter P is simply 5s.

**Also Read: ****Regular Heptagon: Definition and Formulas**

## Types of Pentagon

A pentagon is a five-sided polygon, and there are several types based on the lengths of their sides and the measures of their angles. Here are the main types of pentagons:

### Regular Pentagon

- All sides are of equal length.
- All interior angles are equal, each measuring 108 degrees.
- It has rotational symmetry of order 5 and five lines of symmetry.

### Irregular Pentagon

- Sides and angles are not necessarily equal.
- There is no specific symmetry.
- Each irregular pentagon can vary greatly in shape.

### Convex Pentagon

- All interior angles are less than 180 degrees.
- The pentagon “bulges” outward, with no inward indentations.

### Concave Pentagon

- At least one interior angle is greater than 180 degrees.
- The pentagon has at least one indentation, giving it a “caved-in” appearance.

### Equilateral Pentagon

- All sides are of equal length.
- The angles are not necessarily equal.
- It is a special type of irregular pentagon unless it also has equal angles (then it’s regular).

### Cyclic Pentagon

- All vertices lie on a single circle (circumcircle).
- Can be regular or irregular.
- The sum of the opposite angles is 180 degrees.

### Simple Pentagon

- Does not intersect itself.
- All sides meet at the vertices, forming a single, unbroken shape.

### Complex Pentagon

- Intersects itself, creating a star-like shape.
- Not a simple polygon.

## Properties of Regular Pentagon with Formulas

A regular pentagon is a polygon with five equal sides and five equal interior angles. Here are the key properties of a regular pentagon, along with the relevant formulas:

**1. Equal Sides and Angles:**

- Each side has the same length, denoted by s.
- Each interior angle measures 108 degrees.

**2. Symmetry**:

- A regular pentagon has five lines of symmetry.
- It has rotational symmetry of order 5, meaning it looks the same after a rotation of 72 degrees.

**3. Diagonals**:

- A regular pentagon has five diagonals.
- The diagonals intersect each other at angles and divide each other into segments that are in the golden ratio, approximately 1.618.

**4. Area**:

- The area A of a regular pentagon with side length s is given by the formula:

Area = ¼ √5(5+2√5)s² |

**5. Perimeter**:

- The perimeter P of a regular pentagon with side length s is P=5s.

**6. Circumcircle**:

- A regular pentagon can be inscribed in a circle (circumcircle) with a radius R. The relationship between the side length s and the circumradius R is:

s = R √5 – √5 / 2 |

- Alternatively, the circumradius R can be expressed as:

R = s/2 √5 + √5 / 2 |

**7. Incircle**:

- A regular pentagon can also have an incircle (a circle that touches all sides), with a radius r known as the inradius. The relationship between the side length s and the inradius r is:

a = s/2 cot π/5 |

- Alternatively, the inradius r can be expressed as:

r = s/2 √5 + 2√5 / 5 |

**Also Read: ****Decagon: Definition, Types and Formulas**

## Properties of Regular Pentagon With Solved Examples

These examples explain the essential features of regular pentagons as well as the calculations that are associated with them. By gaining an understanding of these concepts, you will be able to solve a variety of geometric issues and apply this form to real-world situations.

### Example 1: Finding Perimeter and Area

**Given:** Side length = 6 cm, Apothem = 4 cm

**Find:** Perimeter and Area

**Solution:**

- Perimeter = 5 x side length = 5 x 6 cm = 30 cm
- Area = (1/2) x perimeter x apothem = (1/2) x 30 cm x 4 cm = 60 cm²

### Example 2: Finding Side Length from Perimeter

**Given:** Perimeter = 50 cm

**Find:** Side length

**Solution:**

- Perimeter = 5 x side length
- 50 cm = 5 x side length
- Side length = 50 cm / 5 = 10 cm

### Example 3: Finding Apothem from Area and Perimeter

**Given:** Area = 100 cm², Perimeter = 40 cm

**Find:** Apothem

**Solution:**

- Area = (1/2) x perimeter x apothem
- 100 cm² = (1/2) x 40 cm x apothem
- Apothem = (100 cm² x 2) / 40 cm = 5 cm

### Example 4: Finding Interior and Exterior Angle

**Given:** Regular pentagon

**Find:** Measure of each interior and exterior angle

**Solution:**

- Measure of each interior angle = (n-2) x 180 / n = (5-2) x 180 / 5 = 108 degrees
- Measure of each exterior angle = 360 / n = 360 / 5 = 72 degrees

### Example 5: Real-world Application

A traffic sign is shaped like a regular pentagon with a side length of 30 cm. What is the perimeter of the sign?

**Solution:**

- Perimeter = 5 x side length = 5 x 30 cm = 150 cm

## FAQs

**What is a regular pentagon?**

A regular pentagon is a 5-sided shape with equal sides and angles.

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

Area = (1/2) x perimeter x apothem (Where Apothem is the perpendicular distance from the center of the pentagon to the midpoint of a side).

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

The sum of the interior angles of a regular pentagon is 540 degrees.

**What is the measure of each exterior angle of a regular pentagon?**

Each exterior angle of a regular pentagon measures 72 degrees.

**What is the difference between a regular pentagon and a regular hexagon?**

A regular pentagon has 5 sides, while a regular hexagon has 6 sides. Both have equal side lengths and equal interior angles, but the measures of these angles differ.

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