Octagon

Octagon

Irregular octagon with eight sides, shown as an uneven polygon with indentations, representing a non-regular octagon example.

Octagon

A regular octagon (8-sided polygon) with equal sides and equal interior angles, shown as a geometric example of a regular polygon.

Regular Octagon

Key Formula

S = (n - 2) \times 180^\circ = (8 - 2) \times 180^\circ = 1080^\circ

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

Where:

$S$ = Sum of the interior angles of an octagon
$n$ = Number of sides (8 for an octagon)
$A$ = Area of a regular octagon
$s$ = Side length of the regular octagon

Worked Example

Problem: A regular octagon has a side length of 5 cm. Find (a) each interior angle and (b) the area.

Step 1: Find the sum of interior angles using the polygon angle formula with n = 8.

S = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

Step 2: Since a regular octagon has all equal angles, divide the total by 8 to find one interior angle.

\text{Each angle} = \frac{1080^\circ}{8} = 135^\circ

Step 3: Use the regular octagon area formula with s = 5 cm.

A = 2(1 + \sqrt{2})\,(5)^2 = 2(1 + 1.4142)(25)

Step 4: Evaluate the expression.

A = 2(2.4142)(25) = 50 \times 2.4142 \approx 120.71 \text{ cm}^2

Answer: Each interior angle is 135°, and the area is approximately 120.71 cm².

Another Example

Problem: How many diagonals does an octagon have?

Step 1: The number of diagonals of any polygon with n sides is given by the formula:

d = \frac{n(n - 3)}{2}

Step 2: Substitute n = 8.

d = \frac{8(8 - 3)}{2} = \frac{8 \times 5}{2} = \frac{40}{2} = 20

Answer: An octagon has 20 diagonals.

Frequently Asked Questions

How many sides does an octagon have?

An octagon has exactly 8 sides. The prefix "octa-" comes from Greek and means eight. A regular octagon has all 8 sides the same length, while an irregular octagon can have sides of different lengths.

What is the interior angle of a regular octagon?

Each interior angle of a regular octagon measures 135°. You can find this by computing the total sum of interior angles, (8 − 2) × 180° = 1080°, and then dividing by 8. Each exterior angle is 180° − 135° = 45°.

Octagon (8 sides) vs. Hexagon (6 sides)

An octagon has 8 sides and an interior angle sum of 1080°, while a hexagon has 6 sides and an interior angle sum of 720°. In a regular hexagon each interior angle is 120°; in a regular octagon each interior angle is 135°. Both are common in tiling patterns, but only regular hexagons can tile a plane by themselves — regular octagons require squares to fill the gaps.

Why It Matters

The regular octagon is one of the most recognizable shapes in everyday life — stop signs worldwide use it. In architecture and design, octagonal floor plans appear in buildings, gazebos, and decorative tiles. Understanding octagon geometry also builds skill with the general polygon angle formula, which applies to any shape with n sides.

Common Mistakes

Mistake: Confusing octagons with hexagons, thinking both have the same number of sides.

Correction: "Hexa-" means six and "octa-" means eight. A hexagon has 6 sides; an octagon has 8. Remembering the Greek prefixes helps keep them straight.

Mistake: Using 180° × n instead of (n − 2) × 180° for the sum of interior angles.

Correction: The correct formula subtracts 2 from n first. For an octagon: (8 − 2) × 180° = 1080°, not 8 × 180° = 1440°.

Related Terms

Polygon — General term for any closed many-sided shape
Side of a Polygon — Each of the 8 line segments forming an octagon
Regular Polygon — A polygon with all sides and angles equal
Hexagon — A 6-sided polygon, often compared to an octagon
Pentagon — A 5-sided polygon in the polygon family
Interior Angle — Each angle inside the octagon measures 135°
Diagonal — An octagon has 20 diagonals