Polygon Calculator
Calculate area, perimeter, diagonals, apothem, and angles for any regular polygon
Input Values
Hexagon
How to Use the Polygon Calculator
- Enter Number of Sides: Input the number of sides (n ≥ 3)
- Select Input Type: Choose whether to calculate from side, area, perimeter, circumradius, or apothem
- Enter Value: Input the known measurement
- View Results: All properties are calculated instantly
- Copy Values: Click the copy button next to any result
Understanding Regular Polygons
A regular polygon is a polygon with all sides equal in length and all interior angles equal. The simplest regular polygon is the equilateral triangle (3 sides), and the most common ones include squares (4), pentagons (5), and hexagons (6).
Common Regular Polygons:
Key Properties:
- All sides are equal in length
- All interior angles are equal
- Has n-fold rotational symmetry
- Has n lines of symmetry
- Can be inscribed in a circle (all vertices touch the circle)
- A circle can be inscribed inside (touching all sides)
Polygon Formulas
Basic Formulas
Radii & Angles
Interior Angles Reference
| Polygon | Sides | Interior Angle | Sum of Angles | Diagonals |
|---|---|---|---|---|
| Triangle | 3 | 60° | 180° | 0 |
| Square | 4 | 90° | 360° | 2 |
| Pentagon | 5 | 108° | 540° | 5 |
| Hexagon | 6 | 120° | 720° | 9 |
| Heptagon | 7 | 128.57° | 900° | 14 |
| Octagon | 8 | 135° | 1080° | 20 |
| Nonagon | 9 | 140° | 1260° | 27 |
| Decagon | 10 | 144° | 1440° | 35 |
| Dodecagon | 12 | 150° | 1800° | 54 |
Practical Applications
Architecture & Design
- Building floor plans (hexagonal, octagonal)
- Stop signs (octagon)
- Window and skylight designs
- Tile patterns and tessellations
- Gazebos and pavilions
Engineering
- Bolt heads and nuts (hexagonal)
- Pipe cross-sections
- Structural supports
- Machine parts
Nature
- Snowflakes (hexagonal)
- Honeycombs (hexagonal)
- Crystal structures
- Starfish (pentagonal)
Games & Recreation
- Gaming dice (various polygons)
- Board game tiles
- Sports fields and courts
- Puzzles and tessellations
Frequently Asked Questions
How do I find the area of a regular polygon?
Use A = (n × s²) / (4 × tan(π/n)) or the simpler formula A = (1/2) × Perimeter × Apothem.
What is the apothem?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It's also the inradius of the polygon.
How do I calculate the interior angle?
Interior angle = ((n-2) × 180°) / n, where n is the number of sides. For example, a hexagon has (6-2)×180/6 = 120° per angle.
Which polygons can tessellate a plane?
Only three regular polygons can tile a plane without gaps: equilateral triangles, squares, and regular hexagons.