Area of a Pentagon: Formula, Definition, and Examples

The region that is surrounded by a pentagon's five sides is known as the area of the pentagon. Various techniques can be used to compute it depending on the dimensions that are known. The type of pentagon is another factor. The area of a pentagon is represented in square units. A pentagon's area is calculated using its sides and apothem length.

What Is A Pentagon?

A pentagon is a five-sided polygon and a two-dimensional geometrical shape. Its shape could be basic or self-intersecting. In the Pentagon, there are five equal angles. The Greek words "Penta" (which means "five") and "gon" (which means "angles") are the origin of its name.

Apothem - A pentagon's apothem is a line segment traced perpendicular to one of its sides from the pentagon's center.

The distance between a pentagon's center to one of its vertices is known as the radius.

The Formula For Calculating The Area Of A Pentagon

The formula for finding the area of a regular pentagon that is frequently employed is,

Area of pentagon = ½ * p * a

Here, the pentagon's perimeter is represented by "p," and its apothem by "a."

By breaking up an irregular pentagon into smaller polygons, the area of the irregular pentagon can be computed. Following that, the areas of these polygons are calculated and summed to determine the pentagon's area.

If a pentagon's side length is all that is specified, then,

Area of pentagon = \frac{5s^{2}}{4\tan36^{o}}

If only the pentagon's radius is specified, then,

Area of pentagon = \frac{5}{2}r^{2}\sin72^{o}

Types Of Pentagons

Pentagons can be convex or concave, regular or irregular.

Regular pentagon - The sides and angles of a regular pentagon are all equal. It has 72-degree exterior angles and 108-degree interior angles. A regular pentagon's internal angles add up to 540 degrees.

Irregular pentagon - There are no specified angles in an irregular pentagon since it lacks equal sides and/or angles.

Convex pentagon - A pentagon is said to be convex if its vertices, or points, where the sides meet, point outward. All interior angles within a convex pentagon are smaller than 180 degrees.

Concave pentagon - A pentagon is said to be concave if its vertices, or points, where the sides meet, point inward. One or more of the angles of a concave pentagon is more than 180 degrees, that is, a reflex angle.

Perimeter Of Pentagon

We are aware that the lengths of all the sides make up a pentagon's perimeter. If it is a regular pentagon, the lengths of its five sides are all equal. Consider a regular pentagon with a side length of 'a'. In this instance, the formula can be used to determine the perimeter.

Perimeter of pentagon = a+a+a+a+a = 5a

Properties Of Pentagon

The following are some of the pentagon's characteristics:

• The interior angles of the pentagon add up to 540 degrees.

• It is a regular pentagon if all the sides are equal and all the angles are of equal size. It is irregular if not.

• Each interior angle in the regular pentagon is 108°, while each exterior angle is 72°.

• Five equal sides make up an equilateral pentagon.

• A rectangular pentagon's interior angles add up to 540°.