Regular Octagon — Definition, Formula & Examples

A regular octagon is an eight-sided polygon where every side has the same length and every interior angle has the same measure. Each interior angle of a regular octagon is 135°.

A regular octagon is an equilateral and equiangular polygon with exactly eight sides. Its interior angles each measure $\frac{(8-2) \times 180°}{8} = 135°$ , and the sum of all interior angles is $1080°$ .

Key Formula

A = 2(1 + \sqrt{2})\, s^2

Where:

$A$ = Area of the regular octagon
$s$ = Length of one side

How It Works

To find the interior angle of a regular octagon, use the general polygon formula with

n = 8

. The sum of interior angles is

(8 - 2) \times 180° = 1080°

, so each angle is

1080° \div 8 = 135°

. To find the area, you can use the side length

s

along with the apothem (the distance from the center to the midpoint of a side). Stop signs are the most recognizable real-world example of a regular octagon.

Worked Example

Problem: Find the area of a regular octagon with a side length of 5 cm.

Write the formula: Use the area formula for a regular octagon.

A = 2(1 + \sqrt{2})\, s^2

Substitute the side length: Plug in

s = 5

cm.

A = 2(1 + \sqrt{2})(5)^2 = 2(1 + \sqrt{2})(25)

Calculate: Since

\sqrt{2} \approx 1.414

, evaluate the expression.

A = 50(1 + 1.414) = 50(2.414) \approx 120.7 \text{ cm}^2

Answer: The area is approximately

120.7 \text{ cm}^2

Why It Matters

Regular octagons appear in architecture, tiling patterns, and everyday objects like stop signs. Understanding their properties builds a foundation for working with all regular polygons in geometry courses.

Common Mistakes

Mistake: Confusing the interior angle (135°) with the exterior angle (45°).

Correction: The interior and exterior angles at each vertex are supplementary, meaning they add up to 180°. For a regular octagon, the interior angle is 135° and the exterior angle is 45°.