Regular Nonagon Area Calculator
This regular nonagon area calculator works out the area of a regular nonagon, a nine-sided polygon with all sides and all angles equal. Enter the side length, choose your unit, and it returns the area straight away, along with the perimeter and the apothem (the distance from the centre to the middle of a side). A regular nonagon is one of the standard regular polygons, so the area follows a single tidy formula based on the number of sides and the side length. The calculator does the trigonometry for you, so you do not need to look up the cotangent of the central angle by hand. It is unit agnostic: if you enter the side in metres the area comes back in square metres, and if you enter centimetres it comes back in square centimetres. Use it for geometry homework, drafting and design, tiling and paving layouts, or any time you need the area of a regular nine-sided shape quickly. The formula, a worked example and the assumptions are all set out below so you can check the result and understand where it comes from.
The formula
The area of a regular polygon with nine sides is one quarter of the number of sides times the side length squared, times the cotangent of pi divided by the number of sides. In symbols, A = (1/4) n s² cot(pi/n), where n = 9 for a nonagon and s is the side length. The apothem is (s/2) cot(pi/n) and the perimeter is n times s.
Worked example
Take a regular nonagon with a side of 6. The area is (1/4) times 9 times 6 squared times cot(pi/9), the perimeter is 9 times 6 = 54, and the apothem is 3 times cot(pi/9). Enter 6 above to see the exact figures.
Frequently asked questions
What is the area formula for a regular nonagon?
A = one quarter times nine times the side length squared, times the cotangent of pi over 9. The calculator evaluates the cotangent for you.
Does this work only for a regular nonagon?
Yes. It assumes a regular nonagon, meaning all nine sides and all angles are equal. An irregular nonagon needs a different method, such as splitting it into triangles.
What units does it use?
Whatever unit you enter. A side in metres gives an area in square metres, a side in centimetres gives square centimetres. Keep the side in one unit.
Who this calculator is for
This calculator is for students, drafters, designers and anyone who needs the area of a regular nonagon from its side length.
What this calculator assumes
- The polygon is regular, with nine equal sides and equal angles.
- You enter the straight side length, not the apothem or the radius.
- The area is in the square of whatever unit you use for the side.
- Results are rounded for display.
Formula and sources
Related calculators
- Regular Nonagon Perimeter Calculator: perimeter of the same shape.
- Area Calculator: area of common shapes.
- Circle Calculator: area and circumference of a circle.
- Triangle Calculator: triangle area and sides.