Regular Polygon Calculator
Enter the side length of the regular polygon.
Regular Polygon Formulas and Calculations
What is a Regular Polygon?
A regular polygon is a polygon with all sides equal in length and all interior angles equal in measure. Common examples include equilateral triangles, squares, regular pentagons, and regular hexagons.
Key elements of a regular polygon include:
- Sides (n): The number of sides in the polygon
- Side Length (s): The length of each side
- Apothem (a): The perpendicular distance from the center to any side
- Circumradius (R): The distance from the center to any vertex
- Interior Angle: The angle inside the polygon at each vertex
- Central Angle: The angle at the center between two adjacent vertices
Regular Polygon Formulas
Perimeter (P) = n × s (where n is the number of sides and s is the side length)
Area (A) = (P × a) / 2 = (n × s × a) / 2 (where a is the apothem)
Central Angle = 360° / n
Interior Angle = (n - 2) × 180° / n
Apothem (a) = R × cos(π / n) (where R is the circumradius)
Side Length (s) = 2 × R × sin(π / n)
Circumradius (R) = s / (2 × sin(π / n))
How to Use This Calculator
- Enter the number of sides (must be 3 or more).
- Choose whether you know the side length or the apothem.
- Enter the known value and select the unit of measurement.
- Click "Calculate" to see all the polygon properties.
Common Regular Polygons
Triangle (3 sides)
A regular triangle is an equilateral triangle with three equal sides and three equal interior angles of 60°.
Square (4 sides)
A square has four equal sides and four right angles (90°). It is both a regular quadrilateral and a rhombus.
Pentagon (5 sides)
A regular pentagon has five equal sides and five equal interior angles of 108°.
Hexagon (6 sides)
A regular hexagon has six equal sides and six equal interior angles of 120°. Regular hexagons are common in nature, such as in honeycomb structures.
Octagon (8 sides)
A regular octagon has eight equal sides and eight equal interior angles of 135°. Stop signs are typically in the shape of regular octagons.
Applications of Regular Polygons
Practical Uses
- Architecture and Design: Regular polygons are used in floor plans, decorative elements, and structural designs.
- Engineering: Many mechanical parts and structures use regular polygonal shapes.
- Optics: Camera apertures often use regular polygonal shapes.
- Nature: Many natural structures follow regular polygon patterns, like honeycombs (hexagons) and crystals.
- Art and Tessellations: Regular polygons are used in creating geometric patterns and tessellations.
Example Calculations
Example 1: What is the area of a regular hexagon with a side length of 10 cm?
For a hexagon (n = 6) with side length s = 10 cm:
Circumradius (R) = s / (2 × sin(π / n)) = 10 / (2 × sin(π / 6)) = 10 / (2 × 0.5) = 10 cm
Apothem (a) = R × cos(π / n) = 10 × cos(π / 6) = 10 × 0.866 = 8.66 cm
Area = (n × s × a) / 2 = (6 × 10 × 8.66) / 2 = 259.8 cm²
Example 2: What is the perimeter of a regular octagon with an apothem of 12 inches?
For an octagon (n = 8) with apothem a = 12 inches:
Circumradius (R) = a / cos(π / n) = 12 / cos(π / 8) = 12 / 0.9239 = 12.99 inches
Side length (s) = 2 × R × sin(π / n) = 2 × 12.99 × sin(π / 8) = 2 × 12.99 × 0.3827 = 9.94 inches
Perimeter = n × s = 8 × 9.94 = 79.52 inches
FAQ
What is the difference between a regular and irregular polygon?
A regular polygon has all sides of equal length and all interior angles of equal measure. An irregular polygon has sides of different lengths and/or interior angles of different measures.
What is the apothem of a regular polygon?
The apothem is the shortest distance from the center of the regular polygon to any of its sides. It is the perpendicular distance from the center to a side.
How do I find the area of a regular polygon?
The area of a regular polygon can be calculated using the formula: Area = (n × s × a) / 2, where n is the number of sides, s is the side length, and a is the apothem.
What is the sum of interior angles of a polygon?
The sum of all interior angles in any polygon with n sides is (n - 2) × 180°. For a regular polygon, each interior angle equals ((n - 2) × 180°) / n.
Regular Polygon Properties Table
| Polygon | Sides (n) | Interior Angle | Central Angle | Area Formula |
|---|---|---|---|---|
| Triangle | 3 | 60° | 120° | (√3/4) × s² |
| Square | 4 | 90° | 90° | s² |
| Pentagon | 5 | 108° | 72° | (5/4) × s² × cot(π/5) |
| Hexagon | 6 | 120° | 60° | (3√3/2) × s² |
| Heptagon | 7 | 128.57° | 51.43° | (7/4) × s² × cot(π/7) |
| Octagon | 8 | 135° | 45° | 2 × s² × (1 + √2) |