The area of any polygon is given by:

or

where,

*S* is the length of any side

*N* is the number of sides

*π* is PI, approximately 3.142

**NOTE: The area of a polygon that has infinite sides is the same as the area a circle.**

Depending on the information that are given, different formulas can be used to determine the area of a polygon, below is a list of these formulas:

**The length of a side is given:
**

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the area is given by the formula:

**Given the radius (circumradius)
**

If you know the radius (distance from the center to a vertex)**
**

where

*R* is the radius (circumradius)

*N* is the number of sides

*π* is PI, approximately 3.142

*sin* is the sine function calculated in **radians
**

**Given the apothem (inradius)
**

If you know the apothem, or inradius, (the perpendicular distance from center to a side).

*
*

*area = A*^{2}Ntan*(π/N)*

where,

*A* is the length of the apothem (inradius)

*N* is the number of sides

*tan* is the tangent function calculated in **radians**

**Given the ****apothem**** and length of a side
**

If you know the apothem (the perpendicular distance from center to a side) and the length of a side, first determine the perimeter by mutiplying the side length by N. The area is given by:

*area = AP/2
*

where

*A* is the length of the apothem

*P* is the perimeter

**Example 1: ** Find the area of a polygon with the given radius 2 m and the number of sides is 5 using Apothem.

__Solution__**:**

__Step 1:__ Find the apothem.

*Apothem = R * *cos*(**π/N**)*

Apothem = 2 * cos*(3.14 / 5)*

Apothem = 2 * cos*(0.63)*

Apothem = 2 * 0.81

Apothem = 1.62 m

__Step 2:__ Find the area.

*Area = A² * N * *tan*(**π/N**)*

Area = 1.62² * 5 * tan*(3.14 / 5)*

Area = 2.62 * 5 * tan*(0.63)*

Area = 13.1 * 0.73

Area = 9.5 m^{2}

**Example 2: **The heptagon side length is 7.0 cm calculate the area of the heptagon?

__Solution__:

*Area = S*^{2}N / (4tan*(π/N ))*

The side length *S* is 7.0 cm and *N* is the 7 because heptagon has 7 sides, the area can be determined by using the formula below:

*Area = 343 / (4*tan*(π/N ))
*

*Area = 343 / (4*tan*(3.14/7))
*

*Area = 178.18 cm*^{2}

**Example 3: **Calculate the area of a regular polygon with 9 sides and an inradius of 7 cm.

__Solution__:

*Area = A² * N * *tan(*π/N*)*, *where *A* is the inradius

*Area =7*^{2}×9 *× *tan(*π/9*)*
*

*Area = 49* *× 9* *× 0.3639 = 160.4799 cm*^{2}

**
**

**Example 4: **Calculate the area of the polygon with a circumradius of 4 cm and 9 sides.

__Solution:__ Area using the circumradius formula:

*Area = R*^{2}×N×sin(*2π/N*)* / 2*

*Area = 4*^{2}×9×sin(*2π/9*)* / 2
*

*Area =16×9×0.643/2 = 92.59cm*^{2}

**Example 5: ****If the side of a regular hexagon is equal to 4 cm and apothem is equal to the measure of 5 cm. Calculate the measure of the area of a regular hexagon.****
**

__Solution__:

The given regular polygon is hexagon and we know that hexagon has 6 equal sides; each side’s is 4 cm in length; apothem is equal to 5 cm.

The area of the polygon is calculated using the following formula:

*Area = AP/2, *where A is apothem and P is the perimeter of the polygon

The perimeter of the polygon is easily determined by multiply the length of each side with the number of sides, which gives:

*P = 4 * 6 = 24 cm
*

Now we have to substitute the perimeter and apothem values in the above formula, then we have:

*Area = AP/2*

*Area = 5 × 24 / 2 = 60 cm*^{2}

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