Why is the diagonal of a regular pentagon equal to the golden ratio times the side length?

By applying the Law of Cosines to the triangle formed by two adjacent sides of length a and the 108° interior angle: d² = 2a²(1 - cos(108°)) = 2a²(1 + (√5 - 1)/4) = a² · [(3 + √5)/2] = a² · ϕ², where ϕ = (1 + √5)/2 is the Golden Ratio. Therefore, d = a · ϕ ≈ 1.618034a.

How many diagonals does a regular pentagon have?

A regular pentagon has exactly 5 diagonals, and all 5 are congruent with length d = a · ϕ. When all 5 diagonals are drawn, they form a pentagram (five-pointed star) enclosing a smaller inverted regular pentagon in the center.

Polygon Geometry • 5-Sided Golden Polytope

Regular Pentagon Calculator

High-precision bidirectional analytic solver. Enter any single known variable—side length, area, perimeter, apothem (inradius), circumradius, total height, or diagonal—and all other 6 geometric parameters will compute instantaneously with exact golden ratio derivations.

Select Known Input (Driver):

Side Length a Input (Driver)

a = 10

Area A Calculated

cm²

A = (a²/4)·√(25+10√5)

Perimeter P Calculated

P = 5a

Apothem (Inradius) r_i Calculated

r_i = (a/10)·√(25+10√5)

Circumradius R Calculated

R = (a/10)·√(50+10√5)

Diagonal d Calculated

d = a · ϕ = a(1+√5)/2

Total Height H Calculated

H = R + r_i = (a/2)√(5+2√5)

Interactive Analytic Blueprint

Hover vertex for coordinates

Layers:

Side (a)

Apothem (r_i)

Circumradius (R)

Diagonal (d = aϕ)

Height (H)

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

Academic BibTeX Citation:

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

Mathematical Foundations & Golden Ratio Invariants

A regular pentagon is a 5-sided regular polygon with 5-fold rotational symmetry and 5 reflection axes (dihedral group D₅). It holds deep connections to the Golden Ratio (ϕ = (1 + √5) / 2 ≈ 1.618034), the Fibonacci sequence, and Kepler-Poinsot polyhedra.

1 Interior, Exterior & Central Angles

▼

Sum of Interior Angles: For n = 5:

S = (5 - 2) \times 180° = 3 \times 180° = 540° (3π rad)

Individual Interior Angle (α):

α = 540° / 5 = 108° (3π / 5 rad)

Central & Exterior Angle (β):

β = 360° / 5 = 72° (2π / 5 rad)

2 Proof of Golden Ratio Diagonal (d = a · ϕ)

▼

Consider the isosceles triangle formed by two adjacent sides of length a and the 108° interior angle. By the Law of Cosines:

d² = a² + a² - 2a² cos(108°) = 2a²[1 - cos(108°)]

Using the exact trigonometric identity cos(108°) = -cos(72°) = -(√5 - 1) / 4:

1 - cos(108°) = 1 + (\sqrt5 - 1) / 4 = (3 + \sqrt5) / 4

Notice that the square of the Golden Ratio ϕ = (1 + √5) / 2 is:

ϕ² = [(1 + \sqrt5) / 2]² = (6 + 2\sqrt5) / 4 = (3 + \sqrt5) / 2

Therefore 2 · [(3 + √5) / 4] = (3 + √5) / 2 = ϕ². Substituting back:

d² = a² \cdot ϕ² ⟹ d = a \cdot ϕ = a \cdot (1 + \sqrt5) / 2 \approx 1.61803399 \cdot a

3 Apothem (r_i) & Circumradius (R) Exact Radicals

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With half central angle θ = 72° / 2 = 36°:

r i = a / [2 \cdot tan(36°)] = (a / 10) \cdot \sqrt(25 + 10\sqrt5) \approx 0.68819096 \cdot a

R = a / [2 \cdot sin(36°)] = (a / 10) \cdot \sqrt(50 + 10\sqrt5) \approx 0.85065081 \cdot a

4 Area Exact Radical Derivation

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Summing the 5 central triangles with base a and height r_i:

A = 5 \times (½ \cdot a \cdot r i) = (5 / 2)a \cdot [(a / 10)\sqrt(25 + 10\sqrt5)] = (a² / 4) \cdot \sqrt(25 + 10\sqrt5) \approx 1.72047740 \cdot a²

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Frequently Asked Questions

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

The area of a regular pentagon with side length a is A = (a² / 4) · √(25 + 10√5) ≈ 1.720477 · a². If you know the inradius (apothem) r_i, the area is A = 5 · r_i² · √(5 - 2√5) ≈ 3.632712 · r_i².

What is the diagonal length of a regular pentagon? +

The diagonal of a regular pentagon is directly proportional to its side length by the Golden Ratio (ϕ): d = a · ϕ = a · (1 + √5) / 2 ≈ 1.618034 · a.

How many diagonals does a pentagon have? +

A pentagon has exactly 5 diagonals, and all 5 are equal in length. Connecting all 5 diagonals forms a pentagram (five-pointed star) which encloses an inverted smaller regular pentagon in the center.