Area Of A Pentagon Formula

Area of a Pentagon Formula

In Geometry, a pentagon is a polygon with five sides. Its shape could be simple or self-intersecting. The Greek words “Penta” and “gon,” which indicate “five” and “angles,” were combined to create the name. The Pentagon has identical angles at each of its five corners. Every side and angle of a regular pentagon are equal. Pentagons can be convex or concave, regular or irregular. One with equal sides and angles is referred to as a regular pentagon. It has 108 degrees of interior angles and 72 degrees of exterior angles. An irregular pentagon is a figure without defined angles because it lacks equal sides and/or angles.

In contrast to a concave pentagon, which has vertices that point inside, a convex pentagon has vertices that point outward where the sides meet. In a straightforward pentagon, the interior angles add up to 540°. A pentagram is a self-intersecting regular pentagon (sometimes known as a star pentagon). A pentagon’s area is calculated using its sides and apothem length. A polygon with five equal-length sides is called an equilateral pentagon. It can, however, form a family of pentagons since its five internal angles can take a variety of sets of values. On the contrary, since it is equilateral and equiangular, the regular pentagon is distinct up to resemblance (its five angles are equal). A cyclic pentagon is one in which the circumcircle, a circle, passes through each of its five vertices. A sample of a cyclic pentagon is the conventional pentagon.

Area of a Pentagon

The area of the Pentagon, or any polygon, is the entire amount of space taken up by that geometric shape. Area and perimeter are the most fundamental measurements in geometry after the side. In geometry, we examine two sorts of shapes: flat (2-D shapes) and solid (3-D forms). We can only determine the area of 2-D objects; for 3-D shapes, we must calculate the surface area. Geometry also deals with the properties of these forms, providing standard formulae for calculating areas, perimeters, volumes, and so on.

What is a Pentagon?

The word Pentagon means ‘five angles’ since it is derived from the Greek words “Penta” (five) and “gonia” (angles), hence a Pentagon is a geometrical object with five sides and five angles (interior). A regular pentagon has five identical sides, five internal angles of 108° each, and five lines of reflectional and rotational symmetry.

In addition to the geometric object pentagon, “Pentagon” refers to the headquarters of the United States Department of Defence, which resembles the Pentagon itself. This is one of the world’s biggest office towers.

For regular pentagons, if the side is represented by s and apothem length which is represented in the following diagram, we can calculate the area of the pentagon using the formula:

Area of pentagon = 1/2 × p × a = 5/2 × s × a

Properties of a Pentagon

A pentagon is a two-dimensional shape with five sides and five interior angles, with the following properties:

  • The sum of all interior angles of a pentagon is 540°.
  • For Regular Pentagon:
    • All sides are equal.
    • All interior angles are equal and have a measure of 108°.
    • All exterior angles are also equal and have a measurement of 72°.
    • Regular pentagons have five lines of symmetry which divide the pentagon into congruent parts.
    • Also, regular pentagons have five rotational symmetries as well.
  • It has 5 diagonals meeting at the same point.
  • The ratio of the length of its diagonal to the side of the pentagon is always a golden ratio (1 + √5)/

How to Calculate the Area of a Pentagon.

There are various ways to find the area of the Pentagon, which are explained as follows:

Area of Pentagon with Apothem Length

The area of a pentagon is determined by its side and apothem length. The formula of the area of a pentagon is derived by multiplying any side and apothem length by 5/2. Mathematically the formula is given by

Area of Pentagon(A) = (5/2) s × a


  • s is the side
  • a is apothem length

Area of Regular Pentagon

The area of the pentagon can also be calculated only by using the length. If the side of the regular pentagon is s, then the area of the pentagon can be calculated using the following formula:

Area of Pentagon =\bold{\frac{1}{4}\sqrt{5(5+2\sqrt5)}s^2}

Area of Irregular Pentagon

The area of the Irregular Pentagon can be calculated by splitting the pentagon into small triangles of quadrilaterals (whichever is the most efficient according o the problem) and then calculating their individual areas and adding them together to find the area of the irregular pentagon.

Solved Examples on Area of Pentagon

Problem 1. Find the area of a pentagon with a side of 10 cm and an apothem length of 8 cm.

Solution: Side of pentagon = 10 cm

apothem length = 8 cm

We have,

Area = (5/2) × s × a

⇒ A = (5/2) × 10 × 8

⇒ A = 200 cm2

Problem 2. Find the area of a pentagon with a side of 16 cm and an apothem length of 3 cm.

Solution: Side of pentagon = 16 cm

apothem length = 3 cm

We have,

Area = (5/2) × s × a

⇒ A = (5/2) × 16 × 3

⇒ A = 120 cm2

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FAQs (Frequently Asked Questions)

1. What is the Area Of A Pentagon Formula?

The Area Of A Pentagon Formula is

Area of the pentagon = 1/2 p a

where, “p” stands for the pentagon’s perimeter, while “a” stands for its apothem.

2. Where to find questions on the Area Of A Pentagon Formula?

Several questions on the Area Of A Pentagon Formula can be found on the Extramarks website and mobile application.

3. How Many Diagonals are there in a Pentagon?

There are 5 digonals in the geometric object pentagon.