A Nonagon (also known as an Enneagon) is a polygon with 9 sides and 9 angles. While not as common as hexagons or octagons, nonagons appear in specialized architecture, coin designs, and geometric art. A Regular Nonagon has all sides equal in length and all interior angles equal to 140°.
This calculator is designed to solve the geometry of a regular nonagon instantly, whether you know the side length, the radius, or the apothem.
Calculator Features
1. Three Calculation Modes
Geometry problems do not always give you the side length. Our tool adapts to your data:
Side Length (s): Standard calculation.
Circumradius (R): Useful if the shape is inscribed in a circle.
Apothem (a): The distance from the center to the midpoint of a side.
2. Automatic property Derivation
Enter one value, and get them all. If you input the side length, the calculator automatically determines the area, perimeter, circumradius, and apothem.
3. Visual Verification
A dynamic SVG drawing updates in real-time, showing the nonagon with a dimension line indicating exactly which measurement you are editing. This prevents confusion between “radius” and “apothem.”
The Formulas
For a regular nonagon with side length $s$, radius $R$, and apothem $a$:
Given Side ($s$):
Area $A = \frac{9}{4} s^2 \cot\left(\frac{\pi}{9}\right) \approx 6.1818 s^2$
Given Radius ($R$):
Area $A = \frac{9}{2} R^2 \sin\left(\frac{2\pi}{9}\right)$
Given Apothem ($a$):
Area $A = 9 a^2 \tan\left(\frac{\pi}{9}\right)$
Real-World Applications
Architecture (Bahá’í Temples)
The number 9 is sacred in the Bahá’í Faith. All Bahá’í Houses of Worship are nonagonal (9-sided) in structure. Architects use these calculations to design the floor plans and dome bases.
Coinage
Some coins are minted as nonagons. For example, the 50-cent coin of Fiji and commemorative coins from other nations use this shape to distinguish them from circular currency.
Polygons in Pipes
In computer graphics and 3D modeling, approximating a circular pipe often involves using a 9-sided or 12-sided prism to balance visual smoothness with polygon count.
Tips for Success
Radius vs. Apothem
Don’t confuse them! The **Radius** goes to a corner (vertex). The **Apothem** goes to the flat middle of a side. For a nonagon, the Radius is always slightly larger than the Apothem ($R > a$).
Precision Matters
Because $\pi/9$ (20°) and $\cot(20°)$ are irrational numbers, rounding early can lead to errors. Our calculator uses high-precision trigonometric functions internally to ensure accuracy.
Frequently Asked Questions (FAQs)
1. What is the interior angle of a nonagon?
For a regular nonagon, every interior angle is exactly $140^\circ$. The sum of all interior angles is $(9-2) \times 180^\circ = 1260^\circ$.
2. Can I calculate the area of an irregular nonagon?
No. This calculator assumes all sides and angles are equal (“Regular”). For an irregular 9-sided shape, you would need to break it down into triangles or use the “Coordinate Method” (Surveyor’s Formula).
3. How many diagonals does a nonagon have?
A nonagon has 27 diagonals. The formula is $n(n-3)/2$, so $9(6)/2 = 27$.
Final Words
Whether you are designing a sacred temple or just finishing a geometry assignment, the Nonagon Area Calculator saves you from the tedium of trigonometric tables. It turns complex 9-sided math into a simple 1-click solution.
