How the area and perimeter are found
A regular polygon has n equal sides and n equal angles. Its perimeter is simply the side length times the number of sides. Its area comes from splitting the polygon into n identical triangles from the center, using the tangent of the angle each triangle occupies.
- n — number of sides (integer, 3 or more)
- s — length of one side
- P — perimeter
- A — area
More detail
Why area needs the tangent function
A regular polygon can be split into n congruent isosceles triangles meeting at its center. Each triangle's central angle is 2π/n, and solving for its area in terms of the side length introduces tan(π/n) in the denominator. As n grows large, tan(π/n) shrinks and the polygon's area approaches that of a circle with the same perimeter.
Quick check. A square (n = 4, side 1) gives area 1 and perimeter 4 — a fast way to sanity-check the calculator before trying other side counts.
Frequently asked questions
What is the area of a regular hexagon with side 2?
Using A = (n·s²) / (4·tan(π/n)) with n = 6 and s = 2: A = (6·4) / (4·tan(30°)) = 24 / 2.3094 ≈ 10.3923. Enter 6 sides and length 2 above to see this computed directly.
How is the perimeter of a regular polygon calculated?
Perimeter is simply the side length multiplied by the number of sides: P = n · s. For example a regular pentagon (n = 5) with side length 10 has a perimeter of 50.
Why does the calculator require at least 3 sides?
A polygon needs at least 3 sides to enclose an area — with fewer than 3 sides no closed shape exists, so the calculator returns "—" for any n below 3 or any non-whole number of sides.
Does this work for irregular polygons?
No. This formula assumes all n sides and all interior angles are equal (a regular polygon). Irregular polygons need a different method, such as the shoelace formula from each vertex's coordinates.