How to Use Octagon Calculator
The Octagon Calculator gives you instant results for all key properties of a regular octagon from a single input.
- Enter the side length — type the value of side s into the input field. The Octagon Calculator recalculates all outputs in real time.
- Set your unit — type a unit label such as cm, m, or inches so the Octagon Calculator displays labelled results.
- Adjust decimal places — choose how many decimal places you want; the Octagon Calculator rounds results accordingly.
The Octagon Calculator shows area, perimeter, circumscribed radius, and inscribed radius simultaneously, each annotated with its formula, so you always have a clear reference.
Formula & Theory — Octagon Calculator
The Octagon Calculator uses the following exact formulas for a regular octagon with side length s:
Area A = 2(1 + √2) × s² ≈ 4.8284 × s²
Perimeter P = 8s
Circumradius R = s / √(2 − √2) ≈ 1.3066 × s
Inradius r = s(1 + √2) / 2 ≈ 1.2071 × s
| Symbol | Meaning |
|---|---|
| s | Side length of the regular octagon |
| A | Enclosed area |
| P | Total perimeter (eight equal sides) |
| R | Circumscribed (circumradius) — center to vertex |
| r | Inscribed (apothem / inradius) — center to side midpoint |
A regular octagon has eight equal sides and eight equal interior angles of 135°. The Octagon Calculator exploits the closed-form expressions above, derived from the general regular polygon formulas using n = 8.
Derivation Notes
The area formula arises from dividing the octagon into triangles from the center. The circumscribed radius comes from the half-angle formula R = s / (2 sin(π/8)), and the inradius from r = s / (2 tan(π/8)). The Octagon Calculator evaluates all of these numerically for you.
Use Cases for Octagon Calculator
The Octagon Calculator is useful across a wide range of applications:
- Stop sign dimensions — stop signs are regular octagons; use the Octagon Calculator to verify or design sign dimensions.
- Architecture & tiling — octagonal floor tiles and decorative patterns require precise area and radius measurements, which the Octagon Calculator delivers instantly.
- Engineering drawing — mechanical parts with octagonal cross-sections (flanges, sockets) need accurate geometry; the Octagon Calculator simplifies the process.
- Puzzle & game design — octagonal grids and tokens are popular in tabletop games; the Octagon Calculator helps size components correctly.
- Education — students studying regular polygons can use the Octagon Calculator to verify formulas and build geometric intuition.
The Octagon Calculator handles all the trigonometry automatically, saving time whether you are working on a classroom exercise or a professional design project.