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# Area of a Pentagon – Formula, Examples

A pentagon is a polygon with five sides and five angles. It can be either regular or irregular. A regular pentagon has five equal sides and five equal angles, while an irregular pentagon has sides and angles of different lengths and measurements.

## What is the Area of Pentagon?

There are two types of pentagons: regular and irregular.

1. Regular pentagon: A regular pentagon has five equal sides and five equal angles, and its internal angles measure 108 degrees each.
2. Irregular pentagon: An irregular pentagon has sides and angles of different lengths and measurements.

To find the area of a regular pentagon, you can use the following formula:

A = (1/4) x √(5(5 + 2√5)) x s²

Where A is the area of the pentagon and s is the length of each side.

For example, if the length of each side of a regular pentagon is 6 cm, then the area of the pentagon would be:

A = (1/4) x √(5(5 + 2√5)) x 6² = (1/4) x √(5(5 + 2√5)) x 36 ≈ 61.937 cm²

Therefore, the area of a regular pentagon with a side length of 6 cm is approximately 61.937 square centimeters.

## Formula to find the sum of interior angles

The sum of the interior angles of a pentagon is 540 degrees, which can be calculated using the formula:

Sum of interior angles = (n-2) x 180

Where n is the number of sides of the polygon.

## Formula to find the measure of each interior angle

The measure of each interior angle of a regular pentagon can be calculated using the formula:

Measure of each interior angle = (n-2) x 180 / n

For a regular pentagon, n is equal to 5. Thus,

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

## Finding area of an irregular pentagon

To find the area of an irregular pentagon, you can divide it into smaller shapes such as triangles, trapezoids, and rectangles, and then add up their areas. This method is known as triangulation.

## Examples on Area of Pentagon

Example 1: Find the area of a regular pentagon with a side length of 8 cm.

Solution: Using the formula for the area of a regular pentagon, we have:

A = (1/4) x √(5(5 + 2√5)) x = (1/4) x √(5(5 + 2√5)) x 8² ≈ 110.111 cm²

Therefore, the area of a regular pentagon with a side length of 8 cm is approximately 110.111 square centimeters.

Example 2: A regular pentagon has an area of 120 cm². Find the length of each side of the pentagon.

Solution: We can use the formula for the area of a regular pentagon to solve for the length of each side:

A = (1/4) x √(5(5 + 2√5)) x

120 = (1/4) x √(5(5 + 2√5)) x

= 120 x 4 / √(5(5 + 2√5))

≈ 153.48

s ≈ √153.48

s ≈ 12.4 cm

Therefore, the length of each side of the pentagon is approximately 12.4 centimeters.

Example 3: An irregular pentagon is divided into two triangles and a trapezoid, with areas of 30 cm², 24 cm², and 36 cm², respectively. Find the area of the irregular pentagon.

Solution: To find the area of the irregular pentagon, we can add up the areas of the smaller shapes:

Area of pentagon = Area of triangle 1 + Area of triangle 2 + Area of trapezoid

Area of pentagon = 30 + 24 + 36 Area of pentagon = 90 cm²

Therefore, the area of the irregular pentagon is 90 square centimeters.