Calculator Inputs
This calculator works for regular polygons only. Choose one known value and the total number of sides.
Example Data Table
| Polygon | Sides | Side Length | Apothem | Perimeter | Area |
|---|---|---|---|---|---|
| Triangle | 3 | 10 | 2.8868 | 30 | 43.3013 |
| Square | 4 | 10 | 5.0000 | 40 | 100.0000 |
| Pentagon | 5 | 10 | 6.8819 | 50 | 172.0477 |
| Hexagon | 6 | 10 | 8.6603 | 60 | 259.8076 |
| Octagon | 8 | 10 | 12.0711 | 80 | 482.8427 |
Formula Used
From side length: a = s / (2 tan(π / n))
Use this when you know one side length s and the number of sides n.
From circumradius: a = R cos(π / n)
This converts the radius from the center to a vertex into the apothem.
From perimeter: a = (P / n) / (2 tan(π / n))
The side length becomes s = P / n, then the standard apothem relation is applied.
From area: a = √(A / (n tan(π / n)))
This comes from A = n a² tan(π / n) for a regular polygon.
The calculator also uses Area = 0.5 × Perimeter × Apothem to verify and present supporting values.
How to Use This Calculator
- Select the total number of sides for your regular polygon.
- Choose the known value type: side length, perimeter, area, or circumradius.
- Enter the measurement value and your preferred unit label.
- Set the decimal precision and, if needed, the graph comparison limit.
- Press Calculate Apothem to view results above the form.
- Use the CSV and PDF buttons to save the current result set.
Frequently Asked Questions
1. What is an apothem?
An apothem is the perpendicular distance from a regular polygon’s center to the midpoint of any side. It is also the radius of the inscribed circle.
2. Does this work for irregular polygons?
No. The formulas here assume every side and interior angle are equal. Irregular polygons need different geometry methods, often based on coordinates or decomposition.
3. Why can I calculate apothem from area?
For a regular polygon, area depends directly on the apothem and side count. Rearranging the regular polygon area equation gives a direct square-root formula for apothem.
4. What is the difference between apothem and circumradius?
The apothem reaches the middle of a side. The circumradius reaches a vertex. For regular polygons, the circumradius is always equal to or larger than the apothem.
5. Why does the apothem increase with more sides?
When side length stays fixed, polygons with more sides become closer to a circle. That pushes the side midpoint farther from the center, increasing the apothem.
6. Can I use centimeters, meters, or inches?
Yes. The calculator is unit-flexible. Enter any consistent unit label, and it will carry that label into length outputs while showing area in square units.
7. Why is area shown in squared units?
Area measures two-dimensional space. If length is entered in meters, area is square meters. That is why the output label changes to squared form.
8. What happens if I enter too few sides?
A regular polygon must have at least three sides. The calculator validates this rule and displays an error message until a valid side count is entered.