Regular octagon area calculator

What is a regular octagon area calculator?

A regular octagon area calculator finds the area enclosed by an eight-sided polygon whose sides and interior angles are all equal. Because every side has the same length and every interior angle is the same, the area depends on a single input: the side length. The calculator applies a closed-form formula, so there is no need to triangulate the shape or sum sectors manually.

This calculator accepts the side length in any common unit of length and returns the area in the matching square unit. Switching the unit selector reconverts the result automatically without retyping the input.

Key concepts

• Regular octagon — a polygon with eight equal sides and eight equal interior angles. Each interior angle measures 135 degrees.
• Side length (s) — the common length of every edge of the octagon.
• Apothem — the perpendicular distance from the centre of the octagon to the midpoint of one of its sides. For a regular octagon, the apothem equals $\frac{s}{2}(1 + \sqrt{2})$.
• Area (A) — the size of the two-dimensional region enclosed by the eight sides.

How does the calculator work?

A regular octagon can be split into eight congruent isoceles triangles that share the centre as a common vertex. Summing the areas of those triangles, or equivalently multiplying the apothem by half the perimeter, gives a simple closed-form expression.

Formula

The constant $2(1 + \sqrt{2})$ is the same for every regular octagon, so the area scales with the square of the side length.

Worked examples

Example 1: side length 1

For $s = 1$:

Example 2: side length 5 cm

For $s = 5$ cm:

Example 3: side length 10 cm

For $s = 10$ cm:

Doubling the side length quadruples the area, as expected from the $s^2$ term.

Example 4: side length 1 m

For $s = 1$ m:

Switching the input unit to metres and the output unit to square metres yields the same constant scaled by the new unit.

Practical uses

• Architecture and tiling — calculating the floor area of octagonal rooms, gazebos, or pavilions, and estimating material for octagonal tile patterns.
• Mechanical design — sizing octagonal flanges, nut faces, and shaft cross-sections where a regular octagon footprint is preferred for symmetry.
• Urban planning — measuring octagonal plazas and traffic islands, including the familiar stop-sign shape.
• Geometry homework — verifying answers when the regular polygon area formula is applied with n = 8.

Notes

• The side length must be positive; a side of zero or a missing value returns no area.
• The formula assumes a perfectly regular octagon. For an irregular octagon, split it into triangles and sum their areas instead.
• If you only have the apothem $a$, the area is $A = 8 \cdot \frac{s \cdot a}{2}$, where $s = 2a(\sqrt{2} - 1)$.
• For other regular polygons, see the regular hexagon area and regular pentagon area calculators.