How to Find the Area of an Octagon

An octagon is a planar polygon with eight straight sides and eight vertices. The determination of the area of an octagon is foundational in problems involving regular polygons, tiling patterns, and classical geometric constructions.

Mathematical Definition and Classification of Octagons

A polygon with exactly eight sides and eight vertices is termed an octagon. If all sides are congruent and all internal angles are equal, the octagon is classified as a regular octagon. Otherwise, it is termed an irregular octagon. For a regular octagon, each interior angle measures $135^{\circ}$ and each exterior angle measures $45^{\circ}$.

Derivation of the Area Formula for a Regular Octagon in Terms of Side Length

Let the side length of a regular octagon be denoted by $a$. The area of the octagon can be determined by decomposing it into eight congruent isosceles triangles, each formed by connecting the center to two adjacent vertices.

The central angle of each such triangle is $\dfrac{360^{\circ}}{8} = 45^{\circ}$.

Consider one such triangle. The two equal sides are radii of the octagon’s circumscribed circle, each of length $R$, and the base is $a$. Let $O$ be the center, $A$ and $B$ be two adjacent vertices. Triangle $OAB$ is isosceles with $\angle AOB = 45^{\circ}$, and $AB = a$.

Express $a$ in terms of $R$. In triangle $OAB$: \[ AB = 2R \sin\left(\dfrac{45^{\circ}}{2}\right) = 2R \sin 22.5^{\circ} \] Thus, \[ a = 2R \sin 22.5^{\circ} \]

The area of triangle $OAB$ is: \[ \Delta = \dfrac{1}{2}R^2 \sin 45^{\circ} \]

There are 8 such triangles in the regular octagon, so the total area is: \[ A = 8 \left(\dfrac{1}{2}R^2 \sin 45^{\circ}\right) = 4R^2 \sin 45^{\circ} \] Using $\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}$, \[ A = 4R^2 \cdot \dfrac{\sqrt{2}}{2} = 2\sqrt{2} R^2 \]

Next, express $R$ in terms of $a$: from $a = 2R \sin 22.5^{\circ}$, \[ R = \dfrac{a}{2 \sin 22.5^{\circ}} \] Substituting this into the previously found area: \[ A = 2\sqrt{2} \left(\dfrac{a}{2\sin 22.5^{\circ}}\right)^2 = 2\sqrt{2} \cdot \dfrac{a^2}{4 \sin^2 22.5^{\circ}} = \dfrac{\sqrt{2}}{2} \cdot \dfrac{a^2}{\sin^2 22.5^{\circ}} \]

Use the exact value $\sin 22.5^{\circ} = \sqrt{\dfrac{2 - \sqrt{2}}{4}}$: \[ \sin^2 22.5^{\circ} = \dfrac{2 - \sqrt{2}}{4} \] Hence: \[ A = \dfrac{\sqrt{2}}{2} \cdot \dfrac{a^2}{\dfrac{2 - \sqrt{2}}{4}} = \dfrac{\sqrt{2}}{2} \cdot \dfrac{4a^2}{2 - \sqrt{2}} = \dfrac{2\sqrt{2}a^2}{2 - \sqrt{2}} \]

This expression can be rationalised and simplified: \[ A = \dfrac{2\sqrt{2}a^2}{2 - \sqrt{2}} \cdot \dfrac{2 + \sqrt{2}}{2 + \sqrt{2}} = \dfrac{2\sqrt{2}a^2 \cdot (2 + \sqrt{2})}{(2)^2 - (\sqrt{2})^2} = \dfrac{2\sqrt{2}a^2 \cdot (2 + \sqrt{2})}{4 - 2} = \dfrac{2\sqrt{2}a^2 \cdot (2 + \sqrt{2})}{2} \] \[ = \sqrt{2}a^2(2 + \sqrt{2}) = 2a^2(1 + \frac{1}{\sqrt{2}}) \] But since $(2 + \sqrt{2}) = 2(1 + \frac{1}{\sqrt{2}})$, we simplify back to standard form: \[ A = 2a^2(1 + \sqrt{2}) \]

Result: The area of a regular octagon of side length $a$ is $A = 2a^2(1 + \sqrt{2})$.

Formula for Area of a Regular Octagon in Terms of Apothem

Let $a$ denote the apothem (the perpendicular distance from the center to any side) and $P$ denote the perimeter ($P = 8s$, where $s$ is the side length). The area is given by:

\[ A = \dfrac{1}{2} \times \text{Perimeter} \times \text{Apothem} = 4aP \] For a regular octagon with side $s$: \[ A = 4a \cdot 2s = 8a^2(\sqrt{2} - 1) \] using the relationship between apothem and side length: $a = \dfrac{s}{2 \tan(\pi/8)}$.

Formula for Area of a Regular Octagon in Terms of Radius

Let $R$ denote the radius of the circumscribed circle. As derived above, for a regular octagon:

\[ A = 2\sqrt{2} R^2 \]

Calculation Example: Area of a Regular Octagon Given Side Length

Given: Side length $a = 14$ units.

Substitution: $A = 2a^2(1 + \sqrt{2}) = 2 \times (14)^2 \times (1 + \sqrt{2})$.

Simplification: $14^2 = 196$.

$A = 2 \times 196 \times (1 + \sqrt{2}) = 392 \times (1 + 1.4142)$

$A = 392 \times 2.4142 = 946.37$

Final result: $A = 946.37$ square units.

Calculation Example: Area of a Regular Octagon Given Radius

Given: Radius $R = 5$ units.

Substitution: $A = 2\sqrt{2} \cdot (5)^2 = 2\sqrt{2} \cdot 25$.

Simplification: $2\sqrt{2} \cdot 25 = 50\sqrt{2} \approx 70.71$

Final result: $A = 70.71$ square units.

Calculation Example: Determining the Side Length from Area

Given: Area $A = 25.54$ square units.

Formula: $A = 2a^2(1 + \sqrt{2})$

Substitution: $25.54 = 2a^2(1 + 1.4142) = 2a^2 \times 2.4142 = 4.8284 a^2$

Simplification: $a^2 = \dfrac{25.54}{4.8284} = 5.293$

$a = \sqrt{5.293} = 2.3$

Final result: $a = 2.3$ units.

Area Calculation Method for Irregular Octagons

For an irregular octagon, the general formula in terms of a single variable cannot be applied. The octagon is subdivided into elementary geometric regions, typically triangles or quadrilaterals, whose areas are calculated individually and then summed to determine the total area. This approach relies on known methods for elementary shapes such as those described in the Area of Triangle Formula.

Connection to Other Geometric Area Formulas

The area formulas for a regular octagon are closely related to those for other regular polygons, such as the Area of Hexagon Formula and Area of Square Formula. A regular octagon may also be interpreted as the region obtained by removing the corners of a square, creating geometrical connections between these shapes.