Hexadecagon Calculator

Dedicated tool for a regular 16‑sided polygon: enter side / circumradius / inradius / area / perimeter – instantly get all properties, with exact formulas and interactive diagram.

Key formulas for n=16: interior angle = 157.5° , central angle = 22.5° ; side s = 2R sin(π/16) ; inradius r = R cos(π/16) ; area A = 4 s² cot(π/16) = 8 R² sin(π/8).

Hexadecagon, cannot be changed
Positive number
Hexadecagon properties (live)
Sides (n)
16
Side (s)
0.390
Perimeter (P)
6.242
Area (A)
3.062
Circumradius (R)
1.000
Inradius (r)
0.981
Interior angle
157.5°
Central angle
22.5°
hexadecagon
circumcircle (R)
incircle (r)
vertices

Regular Hexadecagon – In‑depth Analysis

A regular hexadecagon is a polygon with 16 equal sides and 16 equal angles. The name comes from Greek hexa·deca (sixteen) and gon (angle). Because 16 is a power of two, it can be constructed using only a compass and straightedge – a property that fascinated ancient Greek geometers and Renaissance architects.

? Basic Geometric Properties

  • Interior angle: (16−2)×180° / 16 = 157.5°. Each internal corner is only 22.5° away from a straight line, making the polygon very close to a circle.
  • Exterior angle: 360° / 16 = 22.5°.
  • Central angle: also 22.5° – the angle subtended by each side at the center.
  • Number of diagonals: n(n−3)/2 = 16×13/2 = 104. The longest diagonal equals the diameter of the circumcircle.
  • Symmetry group: Dihedral group D16, with 16 rotational symmetries and 16 reflection axes.
Exact algebraic expressions (n=16):
sin(π/16) = ½√(2−√(2+√2)) , cos(π/16) = ½√(2+√(2+√2))
For unit circumradius (R=1): side s = 2 sin(π/16) ≈ 0.39018, inradius r = cos(π/16) ≈ 0.98079, area A = 8 sin(π/8) ≈ 3.06147.

? Formula Table (n=16)

GivenSide sCircumradius RInradius rArea APerimeter P
side s—s / (2 sin(π/16))s cot(π/16)/24 s² cot(π/16)16 s
circumradius R2R sin(π/16)—R cos(π/16)8R² sin(π/8)32 R sin(π/16)
inradius r2r tan(π/16)r / cos(π/16)—16 r² tan(π/16)32 r tan(π/16)
area A√(A tan(π/16) / 4)√(A / (8 sin(π/8)))√(A / (16 tan(π/16)))—16 s

Note: π/16 = 11.25°. All trigonometric values have closed‑form radicals; the calculator uses high‑precision floating point.

? Compass‑and‑straightedge Construction

Since 16 = 2⁴, a regular hexadecagon is constructible. A classic method: draw a circle and a diameter AB; construct the perpendicular bisector to get points C, D (a square). Bisect angle ∠COB to obtain 45°, then bisect again to obtain 22.5°. Marking successive 22.5° arcs on the circle yields the 16 vertices. Many Renaissance buildings feature hexadecagonal domes precisely because of this relatively simple construction.

? Radical Forms of sin(π/16) and cos(π/16)

Applying the half‑angle formulas repeatedly:
cos(π/8) = √(2+√2)/2, then
cos(π/16) = √(2+√(2+√2)) / 2    sin(π/16) = √(2−√(2+√2)) / 2.
Thus for R=1, side s = √(2−√(2+√2)) – a concise radical expression that illustrates the algebraic nature of 2‑power polygons.

? Appearances in Nature and Human Artifacts

  • Architecture: Some Italian Renaissance baptisteries and mosques use hexadecagonal floor plans to create a sense of harmony.
  • Games & design: Certain board games (e.g., wargames) employ hexadecagonal grids as a variant of hex grids; some commemorative coins (e.g., Chinese zodiac coins) have a hexadecagonal shape.
  • Mathematical art: M. C. Escher’s Circle Limit series implicitly uses the symmetry of a hexadecagon.
  • Ancient astronomy: Some Han dynasty sundials are divided into 16 parts, corresponding to the 24 solar terms plus intermediate markers.

? Numerical Benchmarks (R=1, s=1, A=1)

ReferenceSide sCircumradius RInradius rArea A
R=1 (unit circumradius)0.3901810.980793.06147
s=1 (unit side)12.562922.5136720.1094
A=1 (unit area)0.223030.571540.560771

? Approximation of π and the Circle

The perimeter of a unit‑circumradius hexadecagon is P = 32 sin(π/16) ≈ 6.24289, while the true circumference 2π ≈ 6.28319 – a relative error of only 0.64%. The area A ≈ 3.06147 vs π ≈ 3.14159 (error 2.55%). This shows how a 16‑gon already gives a decent approximation of the circle, and it was indeed an intermediate step used by Archimedes (who pushed to 96 sides for more accurate bounds).

? Star Polygons {16/3}, {16/5}, {16/7}

Like all regular polygons, the hexadecagon gives rise to star polygons by connecting every k‑th vertex. When k and 16 are coprime (k = 3, 5, 7), we obtain the stars {16/3}, {16/5}, {16/7}. They appear in occult symbolism and decorative mosaics, exhibiting 16‑fold rotational symmetry.

❓ Frequently Asked Questions

It has 16 axes of symmetry: 8 through opposite vertices and 8 through midpoints of opposite sides. All are reflection axes.

For a diagonal connecting vertices k steps apart (k=2,…,8), length dk = 2R sin(kπ/16). For example, the shortest (k=2) is d₂ = 2R sin(π/8) = R√(2−√2) ≈ 0.765R; the longest (k=8) is the diameter 2R.

No, because its interior angle 157.5° does not divide 360° evenly. However, it can be part of semi‑regular tilings, e.g., together with octagons and squares (truncated square tiling).

For any regular polygon, A/P = r/2 (half the inradius). For a unit‑circumradius hexadecagon, r≈0.9808, so A/P≈0.4904 – a quick sanity check for calculations.

The interactive calculator on the left lets you explore these properties with your own inputs.