Regular Polygon Calculator
Calculate regular polygon area, perimeter, apothem, circumradius, and all angles. Supports any polygon from triangle to 1000-gon.
Interior = (n-2) x 180 / nHexagon Properties
Area
259.8076 cm^2
Polygon Input
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Quick-start with common scenarios
Polygon Visualization
Dashed circles: outer = circumscribed (through vertices), inner = inscribed (apothem)
Complete Results
Measurements
Sides
6
Hexagon
Side Length
10.0000 cm
Perimeter
60.0000 cm
Area
259.8076 cm^2
Apothem (Inradius)
8.6603 cm
Circumradius
10.0000 cm
Angles
Interior Angle
120.00deg
Exterior Angle
60.00deg
Central Angle
60.00deg
Sum of Interior
720deg
Additional Properties
Number of Diagonals
9
n(n-3)/2
Can Tile Plane?
Yes
Tessellates!
Common Polygons Reference
| Polygon | Sides | Interior deg | Exterior deg | Diagonals | Tiles? |
|---|---|---|---|---|---|
| Triangle | 3 | 60.00 | 120.00 | 0 | Yes |
| Square | 4 | 90.00 | 90.00 | 2 | Yes |
| Pentagon | 5 | 108.00 | 72.00 | 5 | No |
| Hexagon | 6 | 120.00 | 60.00 | 9 | Yes |
| Heptagon | 7 | 128.57 | 51.43 | 14 | No |
| Octagon | 8 | 135.00 | 45.00 | 20 | No |
| Nonagon | 9 | 140.00 | 40.00 | 27 | No |
| Decagon | 10 | 144.00 | 36.00 | 35 | No |
| Dodecagon | 12 | 150.00 | 30.00 | 54 | No |
Regular Polygon Formulas
Angle Formulas
- Interior = (n-2) x 180 / n
- Exterior = 360 / n
- Sum Interior = (n-2) x 180
- Sum Exterior = 360
Size Formulas
- Perimeter = n x a
- Area = (n x a^2)/(4 tan(pi/n))
- Area = (1/2) x P x apothem
- Diagonals = n(n-3)/2
Radius Formulas
- Apothem (inradius) r = a / (2 tan(pi/n))
- Circumradius R = a / (2 sin(pi/n))
- Ratio R/r = 1 / cos(pi/n)
Hexagon
Area
259.8076 cm^2
?How to Calculate Regular Polygon Properties
Regular polygon formulas: Interior angle = (n-2) x 180 / n degrees. Exterior angle = 360 / n degrees. Area = (n x a^2) / (4 x tan(pi/n)), where n is number of sides and a is side length. Perimeter = n x a. The apothem (inradius) = a / (2 x tan(pi/n)). The circumradius = a / (2 x sin(pi/n)).
What is a Regular Polygon?
A regular polygon is a polygon with all sides equal in length and all interior angles equal. It is both equilateral (equal sides) and equiangular (equal angles). Regular polygons are named by the number of sides: triangle (3), quadrilateral/square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), etc. A circle can be thought of as a regular polygon with infinitely many sides.
Key Facts About Regular Polygons
- Interior angle = (n-2) x 180 / n degrees
- Exterior angle = 360 / n degrees
- Sum of interior angles = (n-2) x 180 degrees
- Sum of exterior angles = 360 degrees (always)
- Perimeter = n x side length
- Area = (1/2) x perimeter x apothem
- Apothem (inradius) = a / (2 x tan(pi/n))
- Circumradius = a / (2 x sin(pi/n))
- Central angle = 360 / n degrees
- Equilateral triangle (n=3), square (n=4), pentagon (n=5), hexagon (n=6)
Quick Answer
Regular polygon formulas: Interior angle = (n-2) x 180 / n degrees. Exterior angle = 360 / n degrees. Area = (n x a^2) / (4 x tan(pi/n)), where n is number of sides and a is side length. Perimeter = n x a. The apothem (inradius) = a / (2 x tan(pi/n)). The circumradius = a / (2 x sin(pi/n)).
Polygon Practice Problems
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Practice with 4 problems to test your understanding.
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Frequently Asked Questions
Interior angle = (n-2) x 180 / n degrees, where n is the number of sides. For example, a hexagon (n=6) has interior angle = (6-2) x 180 / 6 = 120 degrees.
The apothem (also called inradius) is the perpendicular distance from the center to the midpoint of any side. It equals the radius of the largest circle that fits inside the polygon. Formula: apothem = side / (2 x tan(pi/n)).
The circumradius is the distance from the center to any vertex (corner). It equals the radius of the smallest circle that contains the entire polygon. Formula: circumradius = side / (2 x sin(pi/n)).
The number of diagonals in a polygon with n sides is n(n-3)/2. For example, a hexagon has 6(6-3)/2 = 9 diagonals.
Only three regular polygons can tile (tessellate) the plane by themselves: equilateral triangles, squares, and regular hexagons. This is because their interior angles evenly divide 360 degrees.
The sum of exterior angles of any convex polygon is always 360 degrees, regardless of the number of sides. For a regular polygon, each exterior angle = 360/n degrees.
Last updated: 2025-01-15
Hexagon Properties
Area
259.8076 cm^2