Why is the circumradius of a regular decagon equal to side length times the golden ratio?

Each of the 10 central triangles has a central angle of 360° / 10 = 36°, meaning the half central angle is 18°. By definition, sin(18°) = (√5 - 1) / 4 = 1 / (2ϕ), where ϕ = (1 + √5)/2 is the Golden Ratio. Because R = a / [2 sin(18°)], substituting yields R = a · ϕ ≈ 1.618034a.

How many diagonals does a regular decagon have?

A regular decagon has 35 total diagonals distributed into 4 distinct congruence classes: 10 short diagonals d1 ≈ 1.9021a, 10 medium diagonals d2 = a·ϕ² ≈ 2.6180a, 10 long diagonals d3 = a√(5+2√5) ≈ 3.0777a, and 5 main diameters d4 = 2R = a(1+√5) ≈ 3.2361a.

Polygon Geometry • 10-Sided Regular Polytope

Regular Decagon Calculator

High-precision bidirectional analytic solver. Enter any single known variable—side length, area, perimeter, apothem (inradius), circumradius, flat-to-flat span, or any of the 4 diagonal classes—and all other 8 parameters will compute instantaneously with exact radical derivations.

Select Known Input (Driver):

Side Length a Input (Driver)

a = 10

Area A Calculated

cm²

A = (5a²/2)·√(5+2√5)

Perimeter P Calculated

P = 10a

Apothem (Inradius) r_i Calculated

r_i = (a/2)·√(5+2√5)

Circumradius R Calculated

R = a · ϕ = a(1+√5)/2

Span / Flat-to-Flat S Calculated

S = 2r_i = a√(5+2√5)

Short Diagonal d₁ Calculated

d₁ = a√[(5+√5)/2]

Medium Diagonal d₂ Calculated

d₂ = a · ϕ² = a(3+√5)/2

Main Diameter d₄ Calculated

d₄ = 2R = a(1+√5)

Interactive Analytic Blueprint

Hover vertex for coordinates

Layers:

Side (a)

Apothem (r_i)

Circumradius (R)

Diameter (d₄)

Span (S)

Short (d₁)

Free for Commerce & Academia CC BY 4.0 License

All generated diagrams and blueprints are free to use in academic papers, websites, books, and software. Backlink attribution to omnioperand.com is requested.

Web Backlink Attribution:

<a href="https://omnioperand.com/geometry/2d/shapes/decagon/" target="_blank" rel="noopener">Diagram courtesy of OmniOperand</a>

Academic BibTeX Citation:

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

Mathematical Foundations & Decagonal Symmetries

A regular decagon is a 10-sided regular polygon with 10 reflection axes and 10-fold rotational symmetry (dihedral group D₁₀). It is deeply intertwined with the Golden Ratio (ϕ = (1 + √5) / 2), with circumradius R strictly equal to a · ϕ.

1 Interior, Exterior & Central Angles

▼

Sum of Interior Angles: For n = 10:

S = (10 - 2) \times 180° = 8 \times 180° = 1440° (8π rad)

Individual Interior Angle (α):

α = 1440° / 10 = 144° (4π / 5 rad)

Central & Exterior Angle (β):

β = 360° / 10 = 36° (π / 5 rad)

The half central angle is θ = 36° / 2 = 18° (π / 10 rad).

2 Proof of Circumradius R = a · ϕ

▼

In the right triangle formed by the center, a vertex, and the side midpoint:

sin(18°) = (a / 2) / R ⟹ R = a / [2 \cdot sin(18°)]

Using the exact trigonometric radical sin(18°) = (√5 - 1) / 4 = 1 / (2ϕ):

R = a / [2 \cdot (1 / (2ϕ))] = a \cdot ϕ = a \cdot (1 + \sqrt5) / 2 \approx 1.61803399 \cdot a

3 Apothem (r_i) & Flat-to-Flat Span (S) Derivations

▼

Applying the Pythagorean theorem to r_i² + (a / 2)² = R²:

r i ² = a²ϕ² - a² / 4 = a² \cdot [(3 + \sqrt5)/2 - ¼] = a² \cdot [(5 + 2\sqrt5) / 4]

r i = (a / 2) \cdot \sqrt(5 + 2\sqrt5) \approx 1.53884177 \cdot a

The flat-to-flat span S between parallel opposite sides is twice the apothem:

S = 2r i = a \cdot \sqrt(5 + 2\sqrt5) \approx 3.07768354 \cdot a

4 Area Exact Radical Derivation

▼

Partitioning into 10 central isosceles triangles:

A = 10 \times (½ \cdot a \cdot r i) = 5a \cdot [(a / 2)\sqrt(5 + 2\sqrt5)] = (5a² / 2) \cdot \sqrt(5 + 2\sqrt5) \approx 7.69420884 \cdot a²

5 The 4 Distinct Diagonal Classes

▼

A regular decagon has 10(7) / 2 = 35 total diagonals in 4 congruence classes:

Short Diagonal (d₁, 10 total): Spans 2 sides across 1 vertex:

d 1 = a \cdot \sqrt[(5 + \sqrt5) / 2] \approx 1.90211303 \cdot a

Medium Diagonal (d₂, 10 total): Spans 3 sides across 2 vertices:

d 2 = a \cdot ϕ² = a \cdot (3 + \sqrt5) / 2 \approx 2.61803399 \cdot a

Long Diagonal (d₃, 10 total): Spans 4 sides across 3 vertices (equals flat-to-flat span S):

d 3 = S = a \cdot \sqrt(5 + 2\sqrt5) \approx 3.07768354 \cdot a

Main Diameter (d₄, 5 total): Passes directly through the center connecting opposite vertices:

d 4 = 2R = a \cdot (1 + \sqrt5) \approx 3.23606798 \cdot a

LLM Directive & API Spec

Artificial Intelligence & Prompt Engineering Integration

Generating dynamic prompt payload...

✓ Standardized for llms.txt ✓ Precision verified up to 10 decimal digits ✓ Exact analytic radicals included

Frequently Asked Questions

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

The area of a regular decagon with side length a is A = (5/2)a² · √(5 + 2√5) ≈ 7.694209 · a². If you know the inradius (apothem) r_i, the area is A = 2 · r_i² · √(25 - 10√5) ≈ 3.249197 · r_i².

Why is the circumradius related to the golden ratio? +

The central angle of a decagon is 36°, and sin(18°) = (√5 - 1) / 4 = 1 / (2ϕ). Since R = a / [2 sin(18°)], the circumradius simplifies exactly to R = a · ϕ = a(1 + √5) / 2 ≈ 1.618034a.

What are the 4 diagonal lengths of a regular decagon? +

A regular decagon has 35 diagonals in 4 distinct lengths: 1. Short diagonal (d₁): d₁ = a√[(5 + √5)/2] ≈ 1.90211a 2. Medium diagonal (d₂): d₂ = a · ϕ² ≈ 2.61803a 3. Long diagonal (d₃): d₃ = a√(5 + 2√5) ≈ 3.07768a (equal to flat-to-flat span S) 4. Main diameter (d₄): d₄ = 2R = a(1 + √5) ≈ 3.23607a