The area of a pentagon is a bit complex compared to lower sided shapes. Area of a regular pentagon = pa/2, where p = perimeter of the pentagon and a = the apothem (straight line from the centre of the pentagon to the side) of the pentagon. The area of irregular pentagon doesn’t have any formulae as such.

In geometry, a pentagon (from the Greek **πέντε pente** which means five and **γωνία gonia** which means angle) is a five-sided **polygon**.

Index

**Categories of Pentagons**

In general pentagons are divided into two categories: Regular or Irregular; Convex or Concave.

**Regular Pentagon**

A regular pentagon is one with all equal sides and equal angles, with each side making an interior angle 72° at the center of pentagon and the angle between two sides (exterior angle) measuring 108°.

**Irregular Pentagon**

An irregular pentagon is a shape that does not have equal sides and/or angles and no specified angles.

**Convex Pentagon**

A convex pentagon is one whose vertices / where the sides meet, is pointing outwards. All the interior angles are less than 180.

**Concave Pentagon**

Concave pentagon whose vertices / where the sides meet points inwards. All the angles are not less than 180.

**Finding Area of Pentagon**

Area of pentagon can be found in 2 ways.

**Area of Pentagon from the Length of Side and Apothem**

This method is for regular pentagons. Besides the side length, we need to find the “**apothem**” of the pentagon.

The apothem is a straight line from the centre of the pentagon to the side, intersecting the side at a 90º right angle.

**Step 1:** Draw five lines from the center of the pentagon to each vertex (corners).

Now we have five equilateral triangles.

**Step 2:** Calculate the area of the triangle using the formula \(\).

**Step 3:** Multiply the area of one triangle into 5 so that we can get the total area of the pentagon.

**Area of Pentagon Using Formula**

Area of a regular pentagon = pa/2 = 5sa/2

where * p* = the perimeter

*= the apothem*

**a***= side length.*

**s****Area of regular pentagon** = (5*s*^{2}) / (4tan(36º)) = **(5 s^{2}) / (4√(5-2√5))**.

**Properties of Pentagon**

- The sum of all the
**internal**angles in a pentagon is 540°(108° *5) - A pentagon can be simple or self-
**intersecting**.

**Examples**

**Question 1.** Find the area of the regular pentagon whose side is 14.6 cm and apothem length is 10cm.

**Solution.** Given that,

Side(s) = 14.6 cm

Apothem(a) = 10 cm

Area(A) = (5 ⁄ 2) × s × a

= (5 ⁄ 2) × 14.6 × 10 cm^{2}

= 365 cm^{2}.

**Question 2.** Shyam was given a Pentagon of which area is 720 units square and with a side of 19 units. Help him in finding the length of the apothem of the Pentagon?

**Solution. **Given that,

Area of the Pentagon(A) = 720 units square.

Length of a side (s) = 19 units

We know that the area of the Pentagon(A) = 5/2 × s × a.

720 = 5/2 × 19 × a.

Apothem(a) = 15.15 units

Therefore, the length of the apothem of the Pentagon whose are is 720 unit square is 15.15 units.

**Question 3.** Find the area of a pentagon of side 10 cm.

**Solution.** Given side of pentagon(s) = 10 cm

Construct a triangle by joining two of the adjacent vertices with the center of a pentagon.

For pentagon height of the triangle and the apothem are same.

The interior angle O = 360º/5 = 72º.

Since the triangle AOB is an isosceles triangle (AO = BO).

In triangle ∆ AOB = 72º + x + x = 180º.

2x = 180º – 72º.

2x = 108º.

x = 108º*/*2 = 54º.

Now by using the trigonometric ratio to get the value of h.

tan A = *h*(*½ AB*)

So, h = (5) × tan 54º = 6.88.

Area of the triangle = ½ s h = ½10 ×(6.88) = 34.5 cm^{2}.

Area of polygon = 5 × Area of each triangle = 5×34.5 = 172 cm^{2} .

Required area of the pentagon with side 10 cm is 172 cm^{2}.

**FAQs**

**How do we find the area of an irregular pentagon?**For finding the area of an irregular polygon we must first separate the *irregular shape into regular polygons*. We then use the *regular polygon area formulas* for finding the area of each of those polygons. At the end we will *add all those areas* together to get the total area of the irregular polygon.

**What are the differences between a pentagon and a regular pentagon?**A *pentagon* has five straight sides, but does not have to be of equal length but whereas a *regular pentagon* has five equal sides and five equal angles.

**Why does the Pentagon have no parallel lines?**In a pentagon the length of each line is equal to the other ones and the angle between two lines is 108°, which gives a sum of inner angles 540°. In this case there are no parallel lines and it has 5 sides.

**How to find area of pentagon?**