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Area of a Polygon


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 = A2Ntan(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 m2

 

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

Solution:

Area = S2N / (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 / (4tan(π/N ))

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

Area = 178.18 cm2

 

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 =72×9 × tan(π/9)

Area = 49 × 9 × 0.3639 = 160.4799 cm2

 

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

Solution: Area using the circumradius formula:

Area = R2×N×sin(2π/N) / 2

Area = 42×9×sin(2π/9) / 2

Area =16×9×0.643/2 = 92.59cm2

 

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 cm2




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