How do you calculate the side length of an octagon from its area?

To find the side length a from area A, divide the area by 2(1 + √2) and take the square root: a = √(A / [2(1 + √2)]) = √(A · (√2 - 1) / 2) ≈ 0.4550898 · √A.

What are the three different diagonal lengths of a regular octagon?

A regular octagon has 20 total diagonals of 3 distinct lengths: (1) Short diagonal d_S = a√(2 + √2) ≈ 1.84776a connecting vertices separated by one vertex; (2) Medium diagonal (Span / Height) S = a(1 + √2) ≈ 2.41421a connecting opposite parallel sides; (3) Long diagonal d_L = a√(4 + 2√2) = 2R ≈ 2.61313a connecting opposite vertices passing through the center.

What is the miter angle for building an octagon frame or gazebo?

Each corner of a regular octagon has an interior angle of 135°. When joining two pieces of wood to form this corner, each piece must be cut at a 22.5° miter angle (90° - 135°/2 = 22.5°).

Polygon Geometry • 8-Sided Regular Polytope

Regelmatige Achthoek Rekenmachine

Q: What is the formula for the area of a regular octagon?

The area of a regular octagon with side length a is given by A = 2(1 + √2)a² ≈ 4.828427a². In terms of the inradius (apothem) r_i, the area is A = 8(√2 - 1)r_i² ≈ 3.313708r_i². In terms of the span (flat-to-flat distance S = 2r_i), the area is A = 2(√2 - 1)S² ≈ 0.828427S².

High-precision bidirectional analytic solver. Enter any single known variable—such as side length, area, perimeter, apothem (inradius), circumradius, or any of the 3 diagonals—and all other 7 geometric parameters will compute instantaneously with exact radical derivations.

Select Known Input (Driver):

Side Length a Input (Driver)

a = 10

Area A Calculated

cm²

A = 2(1+√2)a²

Perimeter P Calculated

P = 8a

Apothem (Inradius) r_i Calculated

r_i = a(1+√2)/2

Circumradius R Calculated

R = a√(4+2√2)/2

Long Diagonal d_L Calculated

d_L = 2R = a√(4+2√2)

Span / Flat-to-Flat S Calculated

S = 2r_i = a(1+√2)

Short Diagonal d_S Calculated

d_S = a√(2+√2)

Interactive Analytic Blueprint

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Layers:

Side (a)

Apothem (r_i)

Circumradius (R)

Long Diag (d_L)

Span (S)

Short Diag (d_S)

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Academic BibTeX Citation:

@misc{omnioperand_octagon, title={Regular Octagon Calculator & Blueprint}, author={OmniOperand}, url={https://omnioperand.com/geometry/2d/shapes/octagon/}, year={2026}}

Mathematical Foundations & Analytical Derivations

A regular octagon is a two-dimensional convex regular polygon with 8 equal sides and 8 equal interior angles. It belongs to the dihedral symmetry group D₈, featuring 8 axes of reflective symmetry and 8-fold rotational symmetry (45° rotational invariance).

1 Interior, Exterior & Central Angles

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Sum of Interior Angles: For any convex n-gon, the sum of interior angles is given by S = (n - 2) × 180°. For an octagon (n = 8):

S = (8 - 2) \times 180° = 6 \times 180° = 1080° (6π rad)

Individual Interior Angle (α): Because the polygon is regular, all 8 angles are congruent:

α = 1080° / 8 = 135° (3π / 4 rad)

Central & Exterior Angle (β): Dividing a full revolution (360°) equally among 8 sectors yields:

β = 360° / 8 = 45° (π / 4 rad)

The half-central angle subtending each right triangle is θ = 45° / 2 = 22.5° (π / 8 rad).

2 Apothem (Inradius) Exact Radical Derivation

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The apothem r_i is the perpendicular distance from the geometric center to the midpoint of any side. In the right triangle formed by the center, a vertex, and the side midpoint:

tan(22.5°) = (a / 2) / r i ⟹ r i = a / [2 \cdot tan(22.5°)]

Using the tangent half-angle trigonometric identity tan(θ / 2) = sin(θ) / [1 + cos(θ)] with θ = 45°:

tan(22.5°) = (\sqrt2 / 2) / (1 + \sqrt2 / 2) = \sqrt2 / (2 + \sqrt2) = [\sqrt2(2 - \sqrt2)] / (4 - 2) = \sqrt2 - 1

Substituting this into the apothem formula and rationalizing the denominator:

r i = a / [2(\sqrt2 - 1)] = [a(\sqrt2 + 1)] / [2(2 - 1)] = [a(1 + \sqrt2)] / 2 \approx 1.20710678 \cdot a

3 Circumradius Exact Radical Derivation

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The circumradius R is the distance from the center to any vertex:

sin(22.5°) = (a / 2) / R ⟹ R = a / [2 \cdot sin(22.5°)]

Applying the sine half-angle identity sin(22.5°) = √[(1 - cos(45°)) / 2] = √[(1 - √2/2) / 2] = √(2 - √2) / 2:

R = a / [2 \cdot (\sqrt(2 - \sqrt2) / 2)] = a / \sqrt(2 - \sqrt2) = [a\sqrt(2 + \sqrt2)] / \sqrt(4 - 2) = [a\sqrt(4 + 2\sqrt2)] / 2 \approx 1.30656296 \cdot a

4 Area Derivation: Four Distinct Proof Methods

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Method 1: Sum of 8 Isosceles Triangles The octagon can be partitioned into 8 congruent isosceles triangles, each having base a and height equal to apothem r_i:

A = 8 \times (½ \cdot a \cdot r i) = 4a \cdot [a(1 + \sqrt2) / 2] = 2(1 + \sqrt2)a² \approx 4.82842712 \cdot a²

Method 2: Bounding Square Subtraction (Carpentry Cut-Off Method) Enclose the octagon in a tight bounding square. Each of the 4 corners is an isosceles right triangle with hypotenuse a. The legs of each corner triangle are x = a / √2. The total side length of the bounding square is:

S = a + 2x = a + 2(a / \sqrt2) = a(1 + \sqrt2)

The area of the square is S² = a²(1 + √2)² = a²(3 + 2√2). The total area of the 4 corner triangles is 4 × (½ x²) = 2(a² / 2) = a². Subtracting the 4 corners:

A = S² - a² = a²(3 + 2\sqrt2) - a² = 2(1 + \sqrt2)a²

Method 3: From Flat-to-Flat Span (S) Because S = a(1 + √2) ⟹ a = S / (1 + √2) = S(√2 - 1):

A = 2(1 + \sqrt2)[S(\sqrt2 - 1)]² = 2(1 + \sqrt2)(3 - 2\sqrt2)S² = 2(\sqrt2 - 1)S² \approx 0.82842712 \cdot S²

Method 4: From Circumradius (R) Using the triangle area formula with central angle β = 45°:

A = 8 \times [½ R² sin(45°)] = 4R² \cdot (\sqrt2 / 2) = 2\sqrt2 \cdot R² \approx 2.82842712 \cdot R²

5 The 3 Distinct Diagonal Classes & Law of Cosines

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An n-gon has n(n - 3) / 2 diagonals. For n = 8, there are exactly 8(5) / 2 = 20 diagonals, grouped into three distinct congruence classes:

Short Diagonal (d_S, 8 total): Connects vertices separated by 1 vertex. In the triangle formed by two adjacent sides of length a and interior angle 135°, apply the Law of Cosines:

d S ² = a² + a² - 2a² cos(135°) = 2a² - 2a²(-\sqrt2 / 2) = a²(2 + \sqrt2)

d S = a \cdot \sqrt(2 + \sqrt2) \approx 1.84775907 \cdot a

Medium Diagonal / Span (S, 8 total): Connects vertices across two intervening sides. This equals the flat-to-flat height:

S = 2r i = a(1 + \sqrt2) \approx 2.41421356 \cdot a

Long Diagonal / Diameter (d_L, 4 total): Connects directly opposite vertices through the polygon center. This is the diameter of the circumcircle:

d L = 2R = a \cdot \sqrt(4 + 2\sqrt2) \approx 2.61312593 \cdot a

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What is the formula for the area of a regular octagon? +

The area of a regular octagon with side length a is A = 2(1 + √2)a² ≈ 4.828427a². If you know the inradius (apothem) r_i, the area is A = 8(√2 - 1)r_i² ≈ 3.313708r_i². If you know the flat-to-flat span S, the area is A = 2(√2 - 1)S² ≈ 0.828427S².

How do you calculate the side length from the area of an octagon? +

To solve for side length a from area A, invert the formula: a = √[A / (2(1 + √2))] = √[A(√2 - 1) / 2] ≈ 0.4550898 · √A.

How do you find the three diagonal lengths of an octagon? +

A regular octagon has three distinct diagonal lengths: 1. Short diagonal (d_S): d_S = a · √(2 + √2) ≈ 1.84776a 2. Medium diagonal / Span (S): S = a(1 + √2) ≈ 2.41421a 3. Long diagonal (d_L): d_L = a · √(4 + 2√2) ≈ 2.61313a

What miter angle should I cut on a miter saw to build an octagon? +

Each interior angle of a regular octagon is 135°. Because two pieces of wood meet at each corner, divide the corner angle in half: 135° / 2 = 67.5°. Since miter saws measure angles relative to a 90° square cut, set your saw to 22.5° (90° - 67.5° = 22.5°).