Octagon Calculator

A regular octagon is an eight-sided polygon with all sides equal in length and all interior angles equal at 135°. It is one of the most symmetric polygons and is the shape of a standard stop sign.

The area is A = 2(1 + √2) × s², where s is the side length. This simplifies to approximately A ≈ 4.8284 × s². You can also use A = 2 × s × a, where a is the apothem.

Each interior angle of a regular octagon is 135°, calculated as (n-2) × 180° / n = (8-2) × 180° / 8 = 1080° / 8 = 135°. The exterior angle is 45°.

A regular octagon has 20 diagonals, calculated using the formula n(n-3)/2 = 8(5)/2 = 20. These include short, medium, and long diagonals of different lengths.

The apothem is the distance from the center to the midpoint of a side: a = s/(2tan(π/8)) ≈ 1.2071 × s. It is used to calculate area: A = ½ × P × a = ½ × 8s × a.

The circumradius is the distance from the center to any vertex: R = s/(2sin(π/8)) ≈ 1.3066 × s. The octagon can be inscribed in a circle of this radius.

A regular octagon can be constructed by cutting equal triangles from the corners of a square. If the square has side length L, each cut removes a triangle with legs of length s(√2-1)/2, where s is the octagon side.

The perimeter is simply P = 8s, where s is the side length. If you know the area, you can find the side: s = √(A / (2(1+√2))) ≈ √(A / 4.8284).

Rearrange the area formula: s = √(A / (2(1+√2))) ≈ √(A / 4.8284). For example, if A = 120.71, then s = √(120.71 / 4.8284) = √25 = 5.

The ratio A/P = (2(1+√2) × s²) / (8s) = s(1+√2)/4 ≈ 0.6036s. This ratio increases with size, meaning larger octagons are more area-efficient per unit perimeter.

A regular octagon has 8 lines of symmetry: 4 passing through opposite vertices and 4 passing through midpoints of opposite sides. This reflects its D₈ dihedral symmetry group.

A regular octagon can be inscribed in a square by cutting equal triangles from each corner. The square side equals s(1 + √2), where s is the octagon side. The octagon area is about 82.8% of the enclosing square.

No, regular octagons alone cannot tile a plane without gaps. However, octagons combined with squares (in a truncated square tiling) can perfectly tessellate, which is why this pattern appears frequently in floor tiles.

The sum of interior angles is (n-2) × 180° = (8-2) × 180° = 1080°. Since all angles are equal in a regular octagon, each angle is 1080° / 8 = 135°.

The area in terms of circumradius R is A = 2√2 × R² ≈ 2.8284 × R². This follows from the general polygon area formula A = ½nR²sin(2π/n).

The shortest diagonal connects vertices separated by one vertex: d₁ = s√(2 + √2) ≈ 1.8478s. There are 8 such diagonals in a regular octagon.

The longest diagonal connects opposite vertices through the center: d₂ = 2R = s√(4 + 2√2) ≈ 2.6131s. There are 4 such diagonals.

Regular octagons appear in Islamic geometric patterns, floor tiles, gazebos, window designs, and dome cross-sections. The octagonal plan was historically used for baptisteries and tower bases.

The area is A = 2√2 × R², derived from the polygon area formula A = ½nR²sin(2π/n) with n = 8.

Use these relationships: s = 2a·tan(π/8), a = s/(2tan(π/8)), R = s/(2sin(π/8)), s = 2R·sin(π/8). The apothem, circumradius, and side are all linked by the octagon's geometry.

The Octagon Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Octagon Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.