Pentagon is a 5 sided polygon and the shape must be closed one.

When all the sides of the polygon are equal we get a regular pentagon. When the sides are not equal then the pentagon is irregular pentagon.
Regular and Irregular Pentagon

The pentagon has interior angles and exterior angles.
The angles formed at the vertex by joining the side segments lying inside the pentagon are called as interior angles.
Angles formed by extension of one of the side segment with adjoining segment and lying outside the enclosed area of the pentagon is called as exterior angle. In a regular pentagon, all the interior angles measure the same.

In the following figures we can see 5 interior angles marked with blue arcs.
The angles on outer side marked with red arcs are exterior angles. The pentagon has five exterior angles.
Angles of Pentagon
We can use the formula to find the sum of interior sides of a pentagon.
Sum of interior angles of a polygon with “n” sides is given by the formula:
S = (n - 2) $\times$ 180
In the pentagon n =5 , so sum = (5 - 2) $\times$ 180
Sum of angles = 3 $\times$ 180 = 540º.
A pentagon has five interior angles. In a regular convex pentagon all interior angles have equal measures.

The measure of interior angle of the regular polynomial can be obtained by using the formula

m = $\frac{Sum\ of\ interior\ angles}{number\ of\ sides}$

= $\frac{(n - 2) \times 180}{n}$

where n: number of sides of the polygon.

To compute the values of a pentagon we plug in n = 5.

m = $\frac{(5 - 2) \times 180}{5}$ = $\frac{540}{5}$

m = 108º.
The angles formed at the vertex in outer area of the pentagon is called as exterior angle. sum of exterior angles a polygon, one at each vertex is always 360º.

We can use the formula $\frac{360^o}{n}$ to find the measure of one exterior angle of a regular convex polygon.

So measure of each exterior angle of a regular pentagon will be $\frac{360^o}{5}$ = 72º.
$\angle 1 + \angle 2 + \angle 3 + \angle 4 + \angle 5$ = 360$^o$
Exterior Angles of Pentagon
The area of pentagon can be determined by applying different methods.

1. When side length and apothem is given.

Area of a Pentagon

Divide the pentagon in five congruent triangles by joining the vertices to the center. Each of these triangle has side of the pentagon as the base and apothem of the pentagon as the height
Let us first determine area of one triangle;

Area = $\frac{1}{2}$ $\times s \times h$

where s = length of each side and h = apothem (perpendicular drawn from the center).

Since we have five such identical triangles, the area of the pentagon is the total area formed by five such triangles.

Area of pentagon = 5 $\times$ area of each triangle

A(pentagon) = 5 $\times$ $\frac{1}{2}$ $\times s \times h$

= 2.5 $\times$ sh

Example: Find the area of a regular convex pentagon with each side measuring 6 inch and with apothem 5 inch.

Solving Pentagon
Solution:
Here, s = 6inch and h = 5 inch

By applying the rule for area we get
A = 2.5 s h
= 2.5 $\times$ 6 $\times$ 5
=75 sq inches.

2. Area of a regular pentagon can also be found by using trigonometry. Divide the pentagon in five identical triangles.

Consider one triangle formed by joining two of the adjacent vertices with the center of a pentagon.

Pentagon Solved Examples

The interior angle O = $\frac{360^o}{5}$ = 72º.

Since the triangle AOB is an isosceles triangle (AO = BO) ≤ A =≤ B = n.

So the measure of angle A = >
In triangle ∆ AOB = 72º + n + n = 180º
2n = 180º - 72º
2n = 108º.

n = $\frac{108^o}{2}$ = 54º.

Now use trigonometric ratio to get the value of h (height of the triangle which is same as the apothem of the pentagon.)

tan A = $\frac{h}{(half\ of\ AB)}$

So h = $\frac{1}{2}$ s $\times$ tan 54º

= 0.3369 s

So area of the triangle = $\frac{1}{2}$ s h = $\frac{1}{2}$ s $\times$(0.3369 s)

=0.16845 $s^2$
So we can represent the area of a regular pentagon only in terms of its side length

= 5 $\times$ Area of each triangle
= 5 $\times$ (0.16845 $s^2$)
= 0.84225 $s^2$

Example:
Find the area of the given regular pentagon whose each side measures 8 cm.

Solution:
Apply the formula for area with side using trigonometry

A = 0.84225 $s^2$
= 0.84225 $\times$ (8)$^2$
= 53.094 sq cm.