A hexagon is a polygon with six straight sides and six angles. In mathematics, determining the area enclosed by a hexagon is a fundamental topic, especially for regular hexagons where all sides and angles are equal. The area calculation relies on geometric properties and can be approached using side length, apothem, or radius.

Mathematical Structure of the Area of a Regular Hexagon

A regular hexagon is defined as a six-sided polygon in which all sides are equal in length, and all internal angles are equal to $120^\circ$. The sum of the interior angles is $720^\circ$. The regular hexagon can be divided into six congruent equilateral triangles by drawing lines from the center to each vertex.

Derivation of the Area of a Regular Hexagon Using Side Length

Let the length of each side of the regular hexagon be $s$. Each of the six congruent equilateral triangles has side length $s$.

The area of an equilateral triangle with side $s$ is given by:

$\text{Area}_{\triangle} = \dfrac{\sqrt{3}}{4} s^2$

There are six such triangles in the hexagon, therefore the total area $A$ is

$A = 6 \times \dfrac{\sqrt{3}}{4} s^2$

$A = \dfrac{6\sqrt{3}}{4} s^2$

$A = \dfrac{3\sqrt{3}}{2} s^2$

Result: The area of a regular hexagon with side length $s$ is $A = \dfrac{3\sqrt{3}}{2} s^2$.

Expression for the Area of a Regular Hexagon Using the Apothem

The apothem ($a$) of a regular hexagon is the perpendicular distance from the center to any of its sides. The perimeter of the hexagon is $P = 6s$.

The area of a regular polygon is given generally by

$\displaystyle A = \dfrac{1}{2} \times P \times a$

For a regular hexagon:

$A = \dfrac{1}{2} \times 6s \times a$

$A = 3sa$

Result: The area of a regular hexagon with side $s$ and apothem $a$ is $A = 3sa$.

Relationship Between Side Length and Apothem of a Regular Hexagon

In a regular hexagon, the apothem $a$ and side length $s$ are related as follows. By drawing a line from the center to the midpoint of a side, a $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle is formed, where $a$ is the length adjacent to the $30^\circ$ angle and $s/2$ is the side opposite the $30^\circ$ angle.

$\tan 30^\circ = \dfrac{(s/2)}{a}$

$\dfrac{1}{\sqrt{3}} = \dfrac{s}{2a}$

$2a = \sqrt{3} s$

$a = \dfrac{\sqrt{3}}{2} s$

Result: The apothem $a$ of a regular hexagon with side length $s$ is $a = \dfrac{\sqrt{3}}{2}s$.

Derivation of the Area in Terms of the Radius (Circumradius)

Let $R$ denote the circumradius, that is, the distance from the center of the hexagon to a vertex. In a regular hexagon, $R = s$.

The area in terms of the radius is:

$A = \dfrac{3\sqrt{3}}{2} R^2$

Result: For a regular hexagon inscribed in a circle of radius $R$, the area is $A = \dfrac{3\sqrt{3}}{2} R^2$.

Area of a Regular Hexagon Using Coordinates

If the coordinates of the vertices of a regular hexagon are given, the area can be calculated using the shoelace formula. This approach is essential for irregular hexagons as well.

Calculation of Area for Irregular Hexagons

For an irregular hexagon (not all sides and angles equal), the area must be calculated by dividing the hexagon into triangles or other polygons whose areas can be computed and summed, or by using coordinate geometry methods such as the shoelace formula.

Worked Examples: Area of a Hexagon

Example 1: Given the side length $s = 10\,\text{cm}$, compute the area of the regular hexagon.

Substitute in the formula:

$A = \dfrac{3\sqrt{3}}{2} \times (10)^2$

$A = \dfrac{3\sqrt{3}}{2} \times 100$

$A = 150\sqrt{3}\,\text{cm}^2$

Final result: $A = 259.81\,\text{cm}^2$ (using $\sqrt{3} \approx 1.732$).

Example 2: If a regular hexagon has apothem $a = 8\,\text{cm}$ and side length $s = 9\,\text{cm}$, calculate its area.

Use the formula:

$A = 3sa = 3 \times 9 \times 8 = 216\,\text{cm}^2$

Final result: $A = 216\,\text{cm}^2$.

Example 3: Given the area of a regular hexagon as $600\sqrt{3}\,\text{units}^2$, determine the side length.

Set $A = \dfrac{3\sqrt{3}}{2} s^2 = 600\sqrt{3}$.

Multiply both sides by $2$:

$3\sqrt{3} s^2 = 1200\sqrt{3}$

Divide both sides by $3\sqrt{3}$:

$s^2 = \dfrac{1200\sqrt{3}}{3\sqrt{3}} = 400$

$s = 20\,\text{units}$

Final result: Side length is $20\,\text{units}$.

Summary of Area of Hexagon Formulas

For a regular hexagon:

$\text{Area} = \dfrac{3\sqrt{3}}{2} s^2$   (using side length)

$\text{Area} = 3sa$   (using apothem and side length)

$\text{Area} = \dfrac{1}{2} \times P \times a$   (using perimeter $P$ and apothem $a$)

$\text{Area} = \dfrac{3\sqrt{3}}{2} R^2$   (using circumradius $R$)

For more details on polygon areas or related formulas, refer to Area Of A Sector Of A Circle Formula.

FAQs on the Area of a Hexagon

Question 1: What is the area of a regular hexagon with radius $R$?

For a regular hexagon, $R = s$, so the area is $A = \dfrac{3\sqrt{3}}{2} R^2$.

Question 2: How do you derive the area formula for a regular hexagon?

Divide the hexagon into six equilateral triangles, find the area of one, multiply by six, and simplify to get $A = \dfrac{3\sqrt{3}}{2} s^2$.

Question 3: Can the area formula $A = \dfrac{3\sqrt{3}}{2} s^2$ be used for any hexagon?

No, it is valid only for regular hexagons. For irregular hexagons, divide into triangles or use coordinate-based methods.

Additional Resources

For further study of polygons and area formulas, consult the following:

Area And Perimeter Formula

Area Of Square Formula

Area Of Isosceles Triangle Formula