Quadratic Equation Solver (ax² + bx + c = 0)
About Polygon Calculator
The Regular & Irregular Polygon Geometry Calculator computes interior/exterior angles, perimeter ($P = n \times s$), area ($A = \frac{1}{2} n s a$), apothem ($a = \frac{s}{2\tan(\pi/n)}$), circumradius ($R$), and Shoelace Polygon Area for irregular $N$-sided coordinate polygons ($n \ge 3$).
How to Use Polygon Calculator
Step 1
Choose Regular Polygon (by side count and length) or Irregular Polygon (by coordinate vertices).
Step 2
Enter polygon dimensions or paste vertex $(x, y)$ coordinate pairs.
Step 3
Review computed area, perimeter, apothem, and interior angle measures.
Step 4
Click "Copy Polygon Specs".
Practical Use Cases for Polygon Calculator
Architectural Gazebo & Hexagonal/Octagonal Paving Design
Calculate apothem dimensions, edge lengths, and floor area for regular pentagons, hexagons, and octagons.
GIS Land Parcel & Irregular Lot Survey Boundary Area
Calculate exact acreage for irregular boundary lots by inputting $(X, Y)$ coordinate vertices into the Gauss Shoelace Formula.
Input & Output Examples
Calculating Regular Octagon ($n=8$) with Side = 5m
Sides ($n$): `8`, Side Length ($s$): `5.0 meters`
Area: `120.71 m²` | Perimeter: `40.0 m` | Apothem: `6.0355 m` | Interior Angle: `135.0°` | Sum of Angles: `1080.0°`
Key Features & Performance
- ✓Regular Polygons: supports any number of sides ($n = 3$ to $n = 1,000$, e.g. Pentagon, Hexagon, Heptagon, Octagon, Decagon, Dodecagon).
- ✓Irregular Polygons: Gauss Shoelace Formula computes exact planar area and perimeter from coordinate vertex lists.
- ✓Computes: Area, Perimeter, Apothem, Incircle/Circumcircle Radius, Interior and Exterior Angles.
- ✓100% Client-Side geometry engine.
- ✓1-Click Copy polygon specifications.
Key Terminology & Definitions
Apothem
The shortest line segment drawn from the center of a regular polygon perpendicular to any of its sides ($a = s / (2\tan(180^\circ/n))$).
Gauss Shoelace Formula
An algorithm for calculating the area of any simple irregular polygon given the Cartesian coordinates of its vertices: $A = \frac{1}{2}|\sum (x_i y_{i+1} - x_{i+1} y_i)|$.