Area, perimeter, diagonal and apothem of a regular octagon
Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 13, 2026
A regular octagon is an eight-sided polygon with all sides and all interior angles equal. Each interior angle is 135°, and the formulas for area, perimeter, diagonal, and apothem only need the side length s. The constant (1 + √2) ≈ 2.4142 shows up everywhere — that's what makes octagons special.
| Name | Formula | Notes |
|---|---|---|
| Area (side length) | A = 2 × (1 + √2) × s² |
s = side length. Numerical form: A ≈ 4.8284 · s². The simplest form when all you know is the side. |
| Perimeter | P = 8 × s |
Eight equal sides — same units as the side length. |
| Long Diagonal (vertex to vertex) | d = s × √(4 + 2√2) |
d ≈ 2.6131 · s. The longest distance across the octagon (through the center, vertex-to-vertex). |
| Short Diagonal | d₂ = s × √(2 + √2) |
d₂ ≈ 1.8478 · s. Vertex to next-but-one vertex (skipping one). |
| Apothem | a = s × (1 + √2) / 2 |
a ≈ 1.2071 · s. The perpendicular distance from the center to the midpoint of any side. |
| Area from Apothem | A = ½ × P × a = 4 × s × a |
Universal regular polygon formula. Equivalent to the explicit form above. |
| Interior Angle | ∠ = (8 − 2) × 180° / 8 = 135° |
From the polygon angle sum formula. Each interior angle is always 135° in a regular octagon. |
| Exterior Angle | ∠ext = 360° / 8 = 45° |
Exterior angle is supplementary to interior: 180° − 135° = 45°. |
| Circumradius | R = s × √(2 + √2) / 2 |
R ≈ 1.3066 · s. Radius of the circle that passes through all 8 vertices. |
| Inradius | r = a = s × (1 + √2) / 2 |
Radius of the inscribed circle. Same as apothem. |
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