# Octagon Calculator

Calculate regular octagon area, perimeter, apothem, and diagonal from side length. Free online octagon calculator with formulas and instant results.

## What this calculates

Calculate all properties of a regular octagon from its side length. This calculator instantly computes the area, perimeter, apothem, and longest diagonal of any regular octagon.

## Inputs

- **Side Length** — min 0 — The length of one side of the regular octagon.

## Outputs

- **Area** — The area of the regular octagon.
- **Perimeter** — The total length of all eight sides.
- **Apothem** — The distance from the center to the midpoint of a side.
- **Longest Diagonal** — The longest diagonal passing through the center.

## Details

A regular octagon has eight equal sides and eight equal interior angles of 135 degrees each. It is one of the most recognizable geometric shapes, most famously used for stop signs worldwide.

Regular Octagon Formulas (side length = s):

- Area: A = 2(1 + sqrt(2)) x s^2, which is approximately 4.8284 x s^2

- Perimeter: P = 8s

- Apothem: a = (s/2)(1 + sqrt(2))

- Longest Diagonal: d = s x sqrt(4 + 2 x sqrt(2))

The area formula can be derived by dividing the octagon into a central square, four rectangles, and four corner triangles, or equivalently into eight isosceles triangles from the center.

Regular octagons appear in architecture (baptisteries, towers), urban planning (intersections), floor tiling (often combined with squares), engineering (nuts and socket heads), and of course traffic signs. The octagon's near-circular shape provides good area-to-perimeter efficiency while being easier to construct than a true circle.

## Frequently Asked Questions

**Q: What is a regular octagon?**

A: A regular octagon is an eight-sided polygon with all sides equal in length and all interior angles equal to 135 degrees. The sum of its interior angles is (8-2) x 180 = 1080 degrees. It has 8 lines of symmetry and 20 diagonals.

**Q: How do I calculate the area of a regular octagon?**

A: For a regular octagon with side length s, use the formula: Area = 2(1 + sqrt(2)) x s^2. For example, if the side length is 5 cm, the area = 2(1 + 1.4142) x 25 = 2 x 2.4142 x 25 = 120.71 square cm.

**Q: How many diagonals does an octagon have?**

A: A regular octagon has 20 diagonals. The formula for diagonals in any polygon is n(n-3)/2, where n is the number of sides. For an octagon: 8(8-3)/2 = 8 x 5 / 2 = 20. These diagonals come in three different lengths.

**Q: Why are stop signs octagonal?**

A: The octagonal shape was chosen for stop signs in 1922 so drivers could recognize it from any direction, including from behind. The more sides a sign has, the more important its message. The octagon is distinctive enough to be recognized instantly, even when faded, snow-covered, or viewed from a non-standard angle.

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Source: https://vastcalc.com/calculators/math/octagon
Category: Math
Last updated: 2026-04-21