Regular Polygon Calculator
Enter the number of sides and side length to calculate perimeter, area, angles, diagonals, apothem, and circumradius.
What is a polygon calculator?
A polygon calculator is a geometry tool that helps you quickly compute the key measurements of a polygon. In this version, we focus on regular polygons, which are shapes where all sides are equal and all angles are equal. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons.
Instead of manually doing trigonometry every time, the calculator gives you immediate values for perimeter, area, angles, and more. That makes it useful for students, teachers, designers, engineers, and anyone working with shape dimensions.
What this calculator gives you
- Perimeter – the full distance around the polygon.
- Area – the amount of 2D space inside the polygon.
- Interior angle – each inside corner angle for a regular polygon.
- Exterior angle – each outside turn angle.
- Sum of interior angles – total of all inside angles combined.
- Number of diagonals – line segments connecting non-adjacent vertices.
- Apothem – distance from center to the midpoint of a side.
- Circumradius – distance from center to a vertex.
Core formulas used
Given:
- n = number of sides
- s = side length
Formulas:
- Perimeter: P = n × s
- Interior angle: (n − 2) × 180° / n
- Exterior angle: 360° / n
- Sum of interior angles: (n − 2) × 180°
- Diagonals: n(n − 3) / 2
- Apothem: a = s / (2 tan(π/n))
- Circumradius: R = s / (2 sin(π/n))
- Area: A = (P × a)/2 = n × s² / (4 tan(π/n))
Example: regular hexagon
Suppose you enter n = 6 and s = 10. The calculator will compute:
- Perimeter = 60
- Interior angle = 120°
- Exterior angle = 60°
- Sum of interior angles = 720°
- Diagonals = 9
- Area ≈ 259.81 square units
This is the same method used in many geometry textbooks, only faster and less error-prone.
Where polygon calculations are used
Education
Students use polygon formulas in middle school, high school, SAT/ACT prep, and introductory college math classes. A calculator like this can verify homework and reinforce geometric intuition.
Architecture and drafting
Floor patterns, structural layouts, and decorative design elements often use repeated regular polygons. Quick area and perimeter calculations help with material estimates.
Computer graphics and game design
Polygons are fundamental in modeling and rendering. While game engines handle complex geometry, understanding polygon properties helps when building procedural assets, collision boundaries, or stylized shapes.
Tips for accurate results
- Use only values where n ≥ 3. A polygon must have at least three sides.
- Keep units consistent (all meters, all inches, etc.).
- Remember this tool assumes a regular polygon, not irregular shapes.
- For engineering work, apply additional tolerance and rounding standards as needed.
Frequently asked questions
Can this calculator handle irregular polygons?
No. It is designed for regular polygons only. Irregular polygons require additional side/angle coordinates.
What happens as the number of sides increases?
As n gets larger, the polygon increasingly resembles a circle. Interior angles approach 180°, and the exterior angle gets very small.
Why are diagonals useful?
Diagonals are useful for triangulation, internal bracing, graphics mesh understanding, and many geometric proofs.
Final thoughts
A good polygon calculator should be fast, clear, and mathematically reliable. Use the tool above whenever you need regular polygon measurements in seconds, whether you are solving geometry problems or planning real-world designs.