Calculate the area, perimeter, apothem, circumradius, inradius, diagonals, and angles of a regular nonagon (9-sided polygon) from side length, area, perimeter, apothem, or circumradius.
A regular nonagon (also called an enneagon) is a polygon with nine equal sides and nine equal interior angles, each measuring exactly 140°. With 27 diagonals and three distinct diagonal lengths, the nonagon is a rich geometric figure that bridges the gap between common polygons like the hexagon and octagon and the perfectly smooth circle.
The area of a regular nonagon with side length s is A = (9/4) × s² × cot(π/9) ≈ 6.182 × s². Like all regular polygons, this formula comes from decomposing the shape into n congruent isosceles triangles radiating from the center. The apothem — the distance from the center to the midpoint of any side — serves as each triangle's height, and equals (s/2) × cot(π/9). The circumradius (center to vertex) is s / (2 sin(π/9)).
In contrast to the hexagon (which tiles the plane perfectly) or the octagon (which appears in stop signs and architectural motifs), the nonagon is less common in everyday life. However, it does appear in coin designs: the two-dollar coin of Barbados, for instance, is a nonagonal shape. In mathematics, the nonagon is notable because — like the heptagon — it cannot be constructed exactly with compass and straightedge alone. The construction requires trisection of an angle, placing it among the so-called "impossible" classical compass-and-straightedge problems.
This calculator supports five input modes: side length, area, perimeter, apothem, or circumradius. It computes all key properties including all three diagonal lengths, every angle, and provides a comparative table of regular polygons from the triangle through the dodecagon.
Nonagon Area & Properties problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter solve from, unit, and it returns area, perimeter, apothem, circumradius in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Area: A = (9/4) × s² × cot(π/9) ≈ 6.182 × s² Perimeter: P = 9s Apothem: a = (s/2) × cot(π/9) Circumradius: R = s / (2 sin(π/9)) Inradius: r = apothem Interior angle: (n−2)×180°/n = 140° Exterior angle: 360°/n = 40° Diagonals: n(n−3)/2 = 27
Result: Area ≈ 618.18 cm², Perimeter = 90 cm, Apothem ≈ 13.74 cm, Circumradius ≈ 14.62 cm
For a regular nonagon with side 10 cm: Area = (9/4) × 100 × cot(π/9) ≈ 618.18 cm². Perimeter = 9 × 10 = 90 cm. Apothem = 5 × cot(π/9) ≈ 13.74 cm. Circumradius = 10 / (2 sin(π/9)) ≈ 14.62 cm. Short diagonal ≈ 17.84 cm, medium ≈ 24.59 cm, long ≈ 28.79 cm. 27 diagonals total.
This nonagon area & properties tool links the entered values (solve from, unit) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Nonagon Area & Properties shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (area, perimeter, apothem, circumradius) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
A nonagon (or enneagon) is a polygon with 9 sides. A regular nonagon has all sides equal and all interior angles equal (140° each).
A = (9/4) × s² × cot(π/9), which simplifies to approximately 6.182 × s², where s is the side length.
A nonagon has 27 diagonals, calculated by n(n − 3)/2 = 9 × 6 / 2 = 27. They come in three distinct lengths.
Each interior angle is (9 − 2) × 180° / 9 = 140°. The exterior angle is 360°/9 = 40°.
They are two names for the same shape. "Nonagon" uses a Latin prefix (nonus = nine), while "enneagon" uses a Greek prefix (ennea = nine). Both are widely accepted.
Nonagons are rare in everyday objects. The Barbados two-dollar coin is nonagonal. Some architectural designs and decorative tiles use 9-fold symmetry.