Why is the circumradius of a regular hexagon equal to its side length?

Each of the 6 triangles formed by the center and two adjacent vertices has a central angle of 360° / 6 = 60°. Because the two radii legs are equal, the base angles are also (180° - 60°) / 2 = 60°, making all 6 central triangles equilateral. Therefore, Circumradius R strictly equals side length a.

What are the two diagonal lengths of a regular hexagon?

A regular hexagon has 9 total diagonals in 2 distinct classes: (1) 6 Short diagonals d_S = a√3 ≈ 1.73205a connecting vertices separated by one vertex; (2) 3 Long diagonals d_L = 2a connecting directly opposite vertices through the center.

Polygon Geometry • 6-Sided Regular Polytope

Regular Hexagon Calculator

High-precision bidirectional analytic solver. Enter any single known variable—side length, area, perimeter, apothem (inradius), circumradius, flat-to-flat span, or diagonal—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 = (3√3 / 2)a²

Perimeter P Calculated

P = 6a

Apothem (Inradius) r_i Calculated

r_i = (a√3) / 2

Circumradius R Calculated

R = a

Long Diagonal d_L Calculated

d_L = 2a = 2R

Span / Flat-to-Flat S Calculated

S = 2r_i = a√3

Short Diagonal d_S Calculated

d_S = a√3 = S

Interactive Analytic Blueprint

Hover vertex for coordinates

Layers:

Side (a)

Apothem (r_i)

Circumradius (R)

Long Diag (d_L)

Span (S)

Short Diag (d_S)

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

Mathematical Foundations & Hexagonal Symmetries

The regular hexagon has 6 congruent sides and 6 congruent interior angles. It belongs to the dihedral symmetry group D₆ with 6 reflection axes and 6-fold rotational symmetry (60° invariance). It is one of only three regular polygons capable of completely tessellating the 2D Euclidean plane without gaps.

1 Interior, Exterior & Central Angles

▼

Sum of Interior Angles: For n = 6:

S = (6 - 2) \times 180° = 4 \times 180° = 720° (4π rad)

Individual Interior Angle (α):

α = 720° / 6 = 120° (2π / 3 rad)

Central & Exterior Angle (β):

β = 360° / 6 = 60° (π / 3 rad)

2 Proof that Circumradius (R) Equals Side Length (a)

▼

Connect the center O to any two adjacent vertices V_1 and V_2. The central angle is ∠V_1OV_2 = 60°. Since both radii legs are equal (OV_1 = OV_2 = R), the triangle ΔOV_1V_2 is isosceles.

\text{Base angles} = (180° - 60°) / 2 = 60°

Because all three angles are 60°, the triangle is equilateral. Therefore:

R = a

3 Apothem (Inradius) Radical Derivation

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The apothem r_i is the altitude of one of the 6 central equilateral triangles:

r i ² + (a / 2)² = a² ⟹ r i ² = a² - a² / 4 = 3a² / 4

r i = (a \cdot \sqrt3) / 2 \approx 0.86602540 \cdot a

4 Area Exact Radical Derivations

▼

A regular hexagon comprises exactly 6 equilateral triangles of side a:

A = 6 \times (a²\sqrt3 / 4) = (3\sqrt3 / 2)a² \approx 2.59807621 \cdot a²

In terms of the flat-to-flat span S = 2r_i = a√3 ⟹ a = S / √3:

A = (3\sqrt3 / 2) \cdot (S / \sqrt3)² = (\sqrt3 / 2)S² \approx 0.86602540 \cdot S²

5 The 2 Distinct Diagonal Classes

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An n-gon has n(n - 3) / 2 diagonals. For n = 6, there are 6(3) / 2 = 9 diagonals in two congruence classes:

Short Diagonal (d_S, 6 total): Connects alternate vertices across 2 sides. Applying the Law of Cosines with interior angle 120°:

d S ² = a² + a² - 2a² cos(120°) = 2a² - 2a²(-½) = 3a²

d S = a\sqrt3 \approx 1.73205081 \cdot a

Long Diagonal (d_L, 3 total): Connects directly opposite vertices through the center. It equals the circumcircle diameter:

d L = 2R = 2a

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Frequently Asked Questions

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

The area of a regular hexagon with side length a is A = (3√3 / 2)a² ≈ 2.598076a². If you know the inradius (apothem) r_i, the area is A = 2√3 · r_i² ≈ 3.464102r_i². If you know the flat-to-flat span S, the area is A = (√3 / 2)S² ≈ 0.866025S².

Why is the circumradius equal to the side length? +

A regular hexagon is formed by 6 congruent equilateral triangles meeting at the center. Because all sides of an equilateral triangle are equal, the distance from the center to any vertex (Circumradius R) is always strictly equal to the outer side length a.

What are the socket and wrench sizes for a hex bolt? +

Wrench and socket sizes measure the flat-to-flat span (S), which equals 2r_i = a√3. For example, a bolt with a side length of 10 mm requires a socket size of 10√3 ≈ 17.32\text{ mm} (typically a standard 17mm or 18mm socket).