Polygon Calculator
Calculate perimeter, area, apothem, radii, angles, and diagonals based on the number of sides and side length. Polygon Calculator helps students, engineers, designers, and anyone solving regular polygon geometry problems.
Highlights
- Calculate perimeter and area for any regular polygon.
- Find apothem and circumradius from side length.
- Determine interior, exterior, and central angles.
- Calculate the sum of interior angles instantly.
- Find the total number of diagonals in a polygon.
Table of Contents
How Polygon Calculator Works?
The Polygon Calculator determines the geometric properties of a regular polygon, meaning every side and every interior angle is equal. It uses the number of sides, n, and the length of one side, s. The number of sides must be an integer from 3 to 1000, while the side length must be positive. Supported side-length units include mm, cm, m, km, in, ft, yd, and mi.
The perimeter is the total length around the polygon. Since all sides have the same length, multiply the side length by the number of sides.
The apothem is the perpendicular distance from the center to the midpoint of a side. The circumradius is the distance from the center to any vertex. Both values come from dividing the polygon into congruent isosceles triangles.
Here, a is the apothem and R is the circumradius. The area can be calculated from half the product of the perimeter and apothem, or directly from the side length and number of sides.
The angle calculations depend only on the number of sides. The interior angle is the angle inside each vertex, the exterior angle is the turning angle between adjacent sides, and the central angle is formed by two adjacent vertices and the polygon center.
The sum of all interior angles and the number of diagonals are calculated with these formulas:
For example, with 6 sides and a side length of 10 cm, the calculator produces the following approximate results:
| Measurement | Calculation | Result |
|---|---|---|
| Perimeter | 6 × 10 | 60 cm |
| Area | 6 × 10² ÷ (4 tan(π ÷ 6)) | 259.81 cm² |
| Apothem | 10 ÷ (2 tan(π ÷ 6)) | 8.66 cm |
| Circumradius | 10 ÷ (2 sin(π ÷ 6)) | 10 cm |
| Interior angle | (6 − 2) × 180 ÷ 6 | 120° |
| Exterior and central angles | 360 ÷ 6 | 60° |
| Sum of interior angles | (6 − 2) × 180 | 720° |
| Number of diagonals | 6 × (6 − 3) ÷ 2 | 9 |
Practical Use Cases
Students can use the calculator to check geometry homework involving regular triangles, squares, pentagons, hexagons, and polygons with many sides. Entering the side count and side length provides a quick way to verify perimeter, area, apothem, radii, angle measures, and diagonal counts.
Engineers can estimate the basic geometry of regular structural layouts, tanks, frames, platforms, and cross-sections. The perimeter, area, apothem, and circumradius are useful starting values when a design is based on equal sides and symmetrical geometry.
Architects and designers can use the results when planning regular polygonal floor plans, facades, decorative patterns, or radial layouts. The central and exterior angles help determine consistent spacing and turning directions around the design.
CAD and technical drawing users can calculate vertex spacing and overall dimensions before constructing a regular polygon in a drawing or modeling program. The circumradius helps place vertices on a circle, while the apothem helps define the distance to each side.
Artists and craft makers can use the perimeter and area for regular polygon templates, tiling patterns, signs, wooden pieces, and paper designs. The diagonal count can also indicate how many nonadjacent connecting lines are possible within the shape.
Accuracy and Limitations
The calculator uses standard trigonometric formulas for ideal regular polygons and provides numerically rounded results, so displayed decimal values may differ slightly from hand calculations. It accepts only whole-number side counts from 3 through 1000 and positive side lengths within the specified input range. The formulas assume that every side and angle is equal; they do not apply directly to irregular polygons, polygons with unequal sides, distorted shapes, or shapes with measured construction errors. Angle outputs are given in degrees, and the number of diagonals is an exact whole-number count.
Quick Examples
Example #1: Hexagon Basics
Calculate the dimensions of a regular hexagon with 10 cm sides.
Example #2: Pentagon Design
Determine the geometry of a regular pentagon using a side length measured in meters.
Example #3: Large Polygon Layout
Analyze a regular 12-sided polygon with each side measured in feet.
