Regular Polygon Calculator
Calculate the area, perimeter, interior angles, and diagonal of any regular polygon.
Regular Hexagon
All calculations run live in your browser using standard geometric formulas.
What Is the Regular Polygon Calculator?
A regular polygon has all sides equal and all interior angles equal. Given any number of sides n ≥ 3 and a side length s, this calculator derives the area, perimeter, angles, inradius, circumradius, longest diagonal, and number of diagonals, plus a live SVG diagram for n ≤ 24.
- ›The inradius (apothem) is the perpendicular distance from center to a side
- ›The circumradius is the distance from center to any vertex
- ›For n=3 (equilateral triangle): A = (√3/4)s²
- ›For n=4 (square): A = s², diagonal = s√2
- ›As n → ∞, polygon approaches a circle: A → πR²
Formula
Key Formulas, Regular n-gon with side s
Area
A = (n·s²) / (4·tan(π/n))
Perimeter
P = n × s
Interior angle
((n−2)×180) / n
Exterior angle
360 / n
Inradius (apothem)
r = s / (2·tan(π/n))
Circumradius
R = s / (2·sin(π/n))
How to Use
- 1Select a preset shape (Triangle, Square, Pentagon…) or type any n ≥ 3
- 2Enter the side length s in any consistent unit (cm, m, inches, etc.)
- 3Click Calculate, the SVG diagram and all 8 properties appear instantly
- 4Use the area/side² ratio to compare shapes of different sizes
- 5For large n (100+), the polygon closely approximates a circle
Example Calculation
Regular hexagon with side = 5 cm:
Area = (6 × 25) / (4 × tan(30°)) = 150 / (4 × 0.5774) = 64.95 cm²
Perimeter = 6 × 5 = 30 cm
Interior angle = (6−2)×180/6 = 120°
Exterior angle = 360/6 = 60°
Inradius = 5 / (2×tan(30°)) = 4.330 cm
Circumradius = 5 / (2×sin(30°)) = 5 cm
Diagonals = 6×(6−3)/2 = 9
Why hexagons in nature?
Hexagons maximize area per unit perimeter among space-filling polygons. Their inradius/circumradius ratio = cos(30°) ≈ 0.866, making them nearly circular and maximizing storage efficiency in honeycomb structures.
Understanding Regular Polygon
Regular Polygon Reference Table
| n | Name | Interior° | Exterior° | Diagonals |
|---|---|---|---|---|
| 3 | Triangle | 60 | 120 | 0 |
| 4 | Square | 90 | 90 | 2 |
| 5 | Pentagon | 108 | 72 | 5 |
| 6 | Hexagon | 120 | 60 | 9 |
| 8 | Octagon | 135 | 45 | 20 |
| 10 | Decagon | 144 | 36 | 35 |
| 12 | Dodecagon | 150 | 30 | 54 |
Frequently Asked Questions
What makes a polygon "regular"?
Regularity requires both equilateral (all sides equal) and equiangular (all angles equal). A rectangle is equiangular but only regular if it is a square.
- ›Triangle: equilateral triangle is the only regular 3-gon
- ›Quadrilateral: only the square is regular (not rectangles or rhombi)
- ›Polygons with 5+ sides: any n can produce a regular n-gon
- ›Regular polygons are always convex for n ≥ 3
What is the apothem (inradius)?
The apothem connects the center to the midpoint of a side at a right angle. Area = (perimeter × apothem) / 2, analogous to A = ½bh for triangles.
- ›For a square with side s: apothem = s/2
- ›For a regular hexagon with side s: apothem = s√3/2 ≈ 0.866s
- ›Apothem < circumradius for all regular polygons
- ›The apothem is the inscribed circle radius (inradius)
How many diagonals does a regular polygon have?
Each vertex connects to n−3 non-adjacent vertices. Dividing by 2 avoids counting each diagonal twice: diagonals = n(n−3)/2.
- ›Triangle (n=3): 0 diagonals, no non-adjacent vertex pairs
- ›Square (n=4): 2 diagonals (the two crossing diagonals)
- ›Pentagon (n=5): 5 diagonals, forming a pentagram
- ›Hexagon (n=6): 9 diagonals, 3 long + 6 short
What is the sum of interior angles?
Any n-gon can be triangulated into (n−2) triangles. Each triangle has 180°, so total = (n−2)×180°. For regular polygons, all angles are equal.
- ›Triangle: (3−2)×180 = 180° → each angle 60°
- ›Square: (4−2)×180 = 360° → each angle 90°
- ›Hexagon: (6−2)×180 = 720° → each angle 120°
- ›Exterior angles always sum to 360° for any convex polygon
Which regular polygons tessellate the plane?
A regular polygon tessellates only when its interior angle divides 360° evenly. Exactly three satisfy this condition.
- ›Triangle (60°): 6 meet at each vertex, triangular grid
- ›Square (90°): 4 meet at each vertex, square grid
- ›Hexagon (120°): 3 meet at each vertex, honeycomb pattern
- ›Pentagon interior angle = 108°: 360°/108° ≈ 3.33, not an integer
What happens as n increases?
For a fixed circumradius R, the area of a regular n-gon = ½nR²sin(2π/n) → πR² as n → ∞. Archimedes used 96-gons to estimate π.
- ›At n=12, inradius/circumradius ≈ 0.966, nearly circular
- ›At n=100, the polygon is visually indistinguishable from a circle
- ›Archimedes used 96-gons to estimate π ≈ 3.14185
- ›Area/s² ratio increases with n, approaching infinity as shape becomes circular
How do I find side length from area?
Rearranging the area formula: s² = 4A·tan(π/n)/n, so s = √(4A·tan(π/n)/n). Once you have s, all other properties follow.
- ›Square example: A=100 → s = √(400·tan(45°)/4) = √100 = 10 ✓
- ›You can also find s from perimeter (s = P/n)
- ›Or from circumradius: s = 2R·sin(π/n)
- ›Or from inradius: s = 2r·tan(π/n)
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