Compute all key properties of a regular octagon: side length (a), circumradius (R), inradius (r), perimeter, area, and three distinct diagonal lengths. Visualize the octagon, its circumcircle, and center on an interactive canvas.
A regular octagon is an eight-sided polygon with equal sides and equal angles (each interior angle = 135°). It is a highly symmetric shape found in nature (columnar basalt), architecture (dome bases, tiles), and everyday objects (stop signs, bolt heads). The octagon can be inscribed in a circle (circumcircle) and contains an inscribed circle (incircle) tangent to all sides.
Fundamental relations (side a):
Circumradius R = a / (2·sin(π/8)) = a / (2·sin 22.5°)
Inradius r = a / (2·tan(π/8)) = a / (2·tan 22.5°)
Area A = 2·(1+√2)·a² ≈ 4.828427·a²
Perimeter P = 8a
The octagon appears in classical architecture: the Baptistery of Florence, the Dome of the Rock in Jerusalem, and many Islamic geometric patterns. Romans used octagonal rooms in baths to distribute heat. In modern engineering, octagonal bolts (hex vs. oct) provide better grip in some applications. The stop sign’s octagonal shape was chosen for easy recognition even when partially obscured. This calculator brings ancient geometry to your screen with precision.
A regular octagon can be divided into 8 congruent isosceles triangles with central angle 45°. The side a subtends an angle of 45° at the center, so chord length a = 2R sin(22.5°). Hence R = a/(2 sin22.5°). The inradius r is the apothem (height of each triangle) = R cos22.5° = a/(2 tan22.5°). Area is 8 times the area of one triangle = 8·(1/2)·a·r = 4a·r = 2(1+√2)a². Diagonals: longest (opposite vertices) = 2R; medium (spanning 3 edges, central angle 135°) = 2R sin(67.5°) = (1+√2)a; short (spanning 2 edges, central angle 90°) = 2R sin45° = √2·R = a·√(2+√2)? Numerically we present verified values.
Values verified against known geometric constants.
| Side a | Circumradius R | Inradius r | Area | Long diagonal | Medium diagonal | Short diagonal |
|---|---|---|---|---|---|---|
| 5 | 6.5328 | 6.0355 | 120.71 | 13.0656 | 12.0711 | 9.2388 |
| 10 | 13.0656 | 12.0711 | 482.84 | 26.1312 | 24.1421 | 18.4776 |
| 15 | 19.5984 | 18.1066 | 1086.4 | 39.1968 | 36.2132 | 27.7164 |
| 20 | 26.1312 | 24.1421 | 1931.4 | 52.2624 | 48.2842 | 36.9552 |
A homeowner plans to build an octagonal gazebo with a side length of 1.5 meters. Using our calculator: a=1.5 → R≈1.960 m, r≈1.811 m, area≈10.86 m². This tells them the total floor area and the radius needed to mark the perimeter on the ground. The long diagonal (≈3.92 m) helps in positioning beams. The medium diagonal (≈3.62 m) is used for cross‑bracing. By visualizing the octagon on canvas, they can verify proportions before construction.
Octagons have three distinct diagonal lengths. The longest (opposite vertices) spans the full width and equals the diameter of the circumcircle. The medium diagonal (connecting vertices with one vertex between them) is often used in star patterns. The short diagonal (connecting vertices with two vertices between) appears in octagonal grids. Our calculator gives all three to aid in truss design, tiling patterns, and decorative layouts.