Regular Polygon Formulas
A regular polygon has all sides of equal length and all interior angles of equal measure. Every regular polygon can be described by just two parameters: the number of sides (n) and any one of: side length (s), circumradius (R), or apothem (a). This calculator accepts any one of those three inputs and derives all six key properties.
Key Formulas
- Area = (n × s²) / (4 × tan(π/n))
- Perimeter = n × s
- Interior angle = (n − 2) × 180° / n
- Exterior angle = 360° / n
- Circumradius R = s / (2 × sin(π/n))
- Apothem a = s / (2 × tan(π/n))
Named Regular Polygons
Regular polygons with 3 to 10 sides have specific names that appear frequently in geometry, architecture, and design. Understanding their properties helps in tiling, construction, and structural engineering.
- Triangle (3): equilateral triangle — interior angles 60°.
- Square (4): interior angles 90°; tiles a plane perfectly.
- Pentagon (5): interior angles 108°; appears in the US Department of Defense building.
- Hexagon (6): interior angles 120°; tiles a plane perfectly — used in honeycombs and tile patterns.
- Octagon (8): interior angles 135°; used in stop sign shapes and floor tile patterns.
- Decagon (10): interior angles 144°.
Circumradius vs Apothem
The circumradius (R) is the radius of the circumscribed circle — the circle that passes through all vertices of the polygon. The apothem (a) is the radius of the inscribed circle — the largest circle that fits inside the polygon, tangent to each side. The apothem is also the perpendicular distance from the centre to any side. The relationship is: a = R × cos(π/n).
Frequently Asked Questions
How do I find the area of a hexagon with a known side length?
For a regular hexagon with side length s, the area formula simplifies to (3√3 / 2) × s². For example, a hexagon with s = 10 has an area of approximately 259.81 square units. The general polygon formula — (n × s²) / (4 × tan(π/n)) — gives the same result and works for any regular polygon.
What is the difference between a regular and irregular polygon?
A regular polygon has all sides equal AND all interior angles equal. An irregular polygon has sides or angles of different lengths or measures. This calculator only handles regular polygons; for irregular polygons you would need to use the shoelace formula or break the shape into triangles.
Can I use this for real-world construction?
Yes. The apothem is particularly useful for construction: it gives the distance from the centre to each wall for structures like octagonal gazebos or hexagonal floor inlays. The circumradius tells you the minimum radius needed to cut the polygon from a circular blank.