Area of a Regular Polygon

The area of a regular polygon can be calculated using a simple formula that combines two key measurements: the semiperimeter (p) and the apothem (a). $$ A = p \cdot a $$

The apothem is the perpendicular distance from the center of the polygon to one of its sides, while the semiperimeter is half the perimeter. Since the perimeter is denoted by P, the semiperimeter is:

$$ p=\frac{P}{2} $$

This formula works for any regular polygon, including pentagons, hexagons, octagons, and decagons.

Why Does the Formula Work?

A regular polygon can be divided into a set of congruent isosceles triangles, each sharing a common vertex at the center of the polygon.

For example, a regular hexagon can be divided into six identical triangles.

The area of a triangle is given by:

$$ \frac{l \cdot h}{2} $$

where l is the base and h is the height.

In a regular polygon, the height of each triangle is exactly the apothem a. Therefore, the area of one triangle is:

$$ \frac{l \cdot a}{2} $$

If the polygon has n sides, it is divided into n congruent triangles. Multiplying the area of one triangle by n gives the total area of the polygon:

$$ A=n \cdot \frac{l \cdot a}{2} $$

Since the perimeter of a regular polygon is equal to the number of sides multiplied by the side length,

$$ P=n \cdot l $$

the formula becomes:

$$ A=\frac{P \cdot a}{2} $$

Because the semiperimeter is half the perimeter,

$$ p=\frac{P}{2} $$

we obtain the most compact version of the formula:

$$ A=p \cdot a $$

This is the form most commonly used in geometry.

Example: Area of a Regular Hexagon

Consider a regular hexagon with:

• side length l=6
• apothem a=5.2

First, calculate the perimeter:

$$ P=6 \cdot 6=36 $$

Next, find the semiperimeter:

$$ p=\frac{36}{2}=18 $$

Now apply the area formula:

$$ A=p \cdot a $$

$$ A=18 \cdot 5.2=93.6 $$

The area of the hexagon is:

$$ A=93.6 $$

Therefore, the hexagon has an area of 93.6 square units.

Alternative Method: You can also calculate the area by finding the area of one of the six congruent triangles. $$ A_t=\frac{6 \cdot 5.2}{2}=15.6 $$ Since the hexagon is made up of six identical triangles: $$ A=15.6 \cdot 6=93.6 $$ The result is exactly the same.

Example: Area of a Regular Pentagon

Now consider a regular pentagon with:

• side length l=2.5
• apothem a=1.72

Note: A regular pentagon can be divided into five congruent isosceles triangles.

Calculate the perimeter:

$$ P=5 \cdot 2.5=12.5 $$

Then find the semiperimeter:

$$ p=\frac{12.5}{2}=6.25 $$

Finally, apply the area formula:

$$ A=p \cdot a $$

$$ A=6.25 \cdot 1.72=10.75 $$

Therefore, the area of the pentagon is 10.75 square units.

Observations and Notes

• Inverse Formulas
  If you know the area and the apothem, you can calculate the semiperimeter using: $$ p=\frac{A}{a} $$ If you know the area and the semiperimeter, you can find the apothem using: $$ a=\frac{A}{p} $$ Since the perimeter is twice the semiperimeter: $$ P=\frac{2A}{a} $$
• The Apothem and the Incircle
  In a regular polygon, the apothem is equal to the radius of the inscribed circle, also known as the incircle. This relationship is often useful when solving geometric problems.
• As the Number of Sides Increases
  A regular polygon with many sides increasingly resembles a circle. As the number of sides approaches infinity, the apothem approaches the radius of the corresponding circle.
• Relationship Between the Apothem and the Circumradius
  The apothem, the circumradius, and the central angles of a regular polygon are closely connected. Using trigonometric relationships, it is possible to calculate one quantity from the others.
• Equivalence Between a Regular Polygon and a Triangle
  The area of a regular polygon is equal to the area of a triangle whose base is the perimeter of the polygon and whose height is the apothem.
  

The area formula for regular polygons is one of the most useful results in elementary geometry because it provides a quick and elegant way to compute the area of any regular polygon using only its perimeter and apothem.