## Pentagon

### A pentagon is a geometric shape that consists of five sides and five angles

A pentagon is a geometric shape that consists of five sides and five angles. It is a polygon, which is a closed figure with straight sides. The word “pentagon” comes from the Greek words “pente,” meaning five, and “gonia,” meaning angle.

Properties of a Pentagon:

1. Sides: A pentagon has five sides, and each side is a line segment connecting two consecutive vertices.

2. Angles: A pentagon has five angles, and each angle is formed by two consecutive sides.

3. Sum of Interior Angles: The sum of the interior angles of a pentagon is always equal to 540 degrees.

4. Exterior Angles: The exterior angles of a pentagon are obtained by extending one side of the pentagon. The sum of the exterior angles is always 360 degrees.

5. Diagonals: A diagonal of a pentagon is a line segment that connects two non-adjacent vertices. A pentagon has five diagonals.

Types of Pentagons:

1. Regular Pentagon: A regular pentagon has all sides and angles of equal length. It is a symmetrical shape with five congruent angles of 108 degrees each.

2. Irregular Pentagon: An irregular pentagon has sides and angles of different lengths. Its angles can vary, but the sum always adds up to 540 degrees.

Formulas:

1. Perimeter: The perimeter of a pentagon can be calculated by adding the lengths of all its sides.

2. Area: The area of a pentagon depends on whether it is a regular or irregular pentagon.

– For a regular pentagon, the area can be calculated using the formula: (1/4) * √(5 * (5 + 2√5)) * s^2, where ‘s’ is the length of any side.

– For an irregular pentagon, the area can be found by dividing it into smaller triangles or by using more complex formulas like the Brahmagupta’s formula.

Applications:

Pentagons are commonly found in various areas, including:

– Architecture: Many buildings and structures incorporate pentagons in their designs, such as the United States Pentagon building.

– Art: Pentagons are used in art and design, including tessellations and patterns.

– Sport: The dimensions of a soccer ball, which is a truncated icosahedron, are based on pentagonal shapes.

– Mathematics: Pentagons play a significant role in geometry and as examples in mathematical problem-solving.

In summary, a pentagon is a five-sided polygon with its unique properties, types, and formulas that define its characteristics.

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