Nonagon Calculator

Enter the side length of a regular nonagon and instantly obtain its circumradius, inradius, area, perimeter, diagonal lengths, and angles. All calculations are based on exact trigonometric formulas.

Unit is only a label; ensure input value matches selected unit.
e.g., 1, 2, 5.5 — results will show the chosen unit.
a = 2
a = 3
a = 5
a = 10
Privacy first: All computations are done locally in your browser. No data leaves your device.

What is a Regular Nonagon?

A regular nonagon (or enneagon) is a nine‑sided polygon with all sides equal and all interior angles equal. Each interior angle measures (9−2)×180°/9 = 140°, and the central angle (subtended by each side at the center) is 360°/9 = 40°. Because 9 is not a Fermat prime, a regular nonagon cannot be constructed exactly using only a compass and straightedge – a famous result proved by Gauss. Nevertheless, it appears frequently in architecture, design, coinage, and mathematical education.

Key formulas for a regular nonagon (side a)

Circumradius: \( R = \frac{a}{2 \sin(\pi/9)} \)
Inradius: \( r = \frac{a}{2 \tan(\pi/9)} \)
Area: \( A = \frac{9}{4} a^2 \cot(\pi/9) \)
Diagonal spanning \( k \) sides: \( d_k = 2R \sin(k\pi/9) \), k = 2,3,4

Historical & Mathematical Significance

The nonagon has intrigued mathematicians since antiquity. Archimedes likely explored approximate constructions. In the 19th century, Gauss proved that a regular n‑gon is constructible by compass and straightedge only if n is a product of a power of two and distinct Fermat primes. Since 9 is not a Fermat prime (the known Fermat primes are 3, 5, 17, 257…), the regular nonagon is not constructible. This discovery linked number theory with geometry in a profound way. Despite this, the nonagon remains a popular shape in decorative art, tiling, and modern structural design.

Why Use a Nonagon Calculator?

  • Instant & accurate – Avoid manual trigonometric calculations; get all dimensions at once.
  • Educational aid – Students can explore how radius and area scale with side length.
  • Practical design – Architects, woodworkers, and jewelers often need precise nonagon measurements for windows, tiles, or medallions.
  • Research reference – Provides reliable numeric values for geometry projects or papers.

Derivation Outline

  1. A regular nonagon can be divided into 9 congruent isosceles triangles with vertex at the center, base = side a, and apex angle = central angle = 40°.
  2. The equal legs are the circumradius R. From the base relation: a = 2R sin(20°) → R = a/(2 sin20°).
  3. The inradius r is the height of each isosceles triangle: r = R cos20° = a/(2 tan20°).
  4. Area = 9 × (area of one triangle) = 9 × (½ a r) = (9/2) a r = (9/4) a² cot20°.
  5. Diagonal lengths correspond to chords subtending multiples of the central angle: dₖ = 2R sin(k·20°).

Example Values (side a = 1, unitless)

Verified with this calculator.

Side a Circumradius R Inradius r Area A Short diag. d₂ Medium diag. d₃ Long diag. d₄
1 1.461902 1.373739 6.181824 1.879385 2.532089 2.879385
2 2.923804 2.747478 24.727296 3.758770 5.064178 5.758770
3 4.385706 4.121217 55.636416 5.638155 7.596267 8.638155
5 7.309510 6.868695 154.545600 9.396925 12.660445 14.396925
10 14.619020 13.737390 618.182400 18.793850 25.320890 28.793850
Application: Nine‑sided Building Dome

A cultural center plans a regular‑nonagon dome with side length 2.5 meters. Using this calculator, the architect obtains circumradius ≈ 3.6548 m, inradius ≈ 3.4343 m, and area ≈ 38.636 m². These figures are essential for estimating the surface area, material quantities, and positioning of support beams. Diagonal lengths also help design internal trusses.

Interesting Fact: Nonagon and Trigonometry

The trigonometric values for 20° (π/9) appear in many identities, but they cannot be expressed using only square roots (unlike 18° or 30°). This is why the nonagon is not constructible. However, approximations like sin20° ≈ 0.3420 are sufficient for practical work.

JavaScript Implementation (core) 

const n = 9;
const angle = Math.PI / n; // 20°
let a = parseFloat(sideInput);
let R = a / (2 * Math.sin(angle));
let r = a / (2 * Math.tan(angle));
let area = (n/4) * a * a * Math.cos(angle) / Math.sin(angle); // cot = cos/sin
let perimeter = n * a;
let d2 = 2 * R * Math.sin(2*angle); // 40°
let d3 = 2 * R * Math.sin(3*angle); // 60°
let d4 = 2 * R * Math.sin(4*angle); // 80°

Common Misconceptions

  • “A regular nonagon can be drawn with compass and straightedge.” – No, Gauss proved it impossible because 9 is not a Fermat prime.
  • “All diagonals have the same length.” – No, there are four distinct diagonal lengths (for k = 2,3,4; k=5 equals k=4 by symmetry).
  • “Area is proportional to side length.” – Area scales with the square of the side, not linearly.
  • “Interior angle is 140°, so exterior is 40° (same as central angle).” – Correct, but the two angles are easily confused.

Applications Across Fields

  • Architecture – Nonagonal floor plans, windows, decorative ceilings.
  • Numismatics – Some commemorative coins have a nonagonal shape.
  • Mathematics education – Classic example for trigonometric functions and polygon properties.
  • Game design – Creating nonagonal grids or ability trees.

Authoritative foundation – This tool is built on Euclidean geometry and trigonometry as presented in standard textbooks (e.g., Coxeter’s “Geometry Revisited”, “CRC Standard Mathematical Tables”). Calculations use double‑precision arithmetic, accurate to 15 decimal digits. Content reviewed by the GetZenQuery Tech team. Last updated March 2026.

Frequently Asked Questions

Yes. Sum of interior angles = (n-2)×180° = 1260° for n=9, so each angle = 1260°/9 = 140°. You can also use exterior angle = 360°/9 = 40°, so interior = 180°−40° = 140°.

Any unit (cm, m, inch, etc.) works; all output values will be in the same unit. You can select a unit from the dropdown, and it will be displayed next to the results.

A regular nonagon has four distinct diagonal lengths corresponding to steps k = 2,3,4 (k=5 equals k=4 because sin(5π/9)=sin(4π/9)). We provide the three shortest distinct ones; the longest (k=4) is listed as “long diagonal”.

Use a protractor to mark 40° central angles, or calculate the circumradius R from the side length and draw a circle with radius R, then step around the circle using the side length as chord distance. Many graphics programs also have polygon tools.

The correct value for d₂ (spanning 2 sides) is 2R sin(40°) ≈ 1.8794. The value 1.7321 corresponds to √3, which is the length of a diagonal in a regular hexagon, not a nonagon. This calculator now uses the correct formula.

The tool uses JavaScript’s double‑precision floating point (≈15‑digit accuracy). Minor discrepancies can arise from rounding in intermediate steps; they are usually negligible for practical purposes.
References: MathWorld Nonagon; Wikipedia: Nonagon; Coxeter, H.S.M. “Regular Polytopes” (Dover, 1973); Gauss, C.F. “Disquisitiones Arithmeticae” (1801).