The area of pentagon can be determined by applying different methods.

1.

When side length and apothem is given.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.

**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.

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.