Regular Polygon Calculator
Calculate area, perimeter, interior angles, and apothem of any regular polygon from the number of sides and side length or circumradius. Includes triangles through dodecagons.
📐 Geometry📐 A = ½n·s·a (apothem method)🔢 Math
Number of sides n
Side length s
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Regular Polygon Calculator | — | A = ½n·s·a (apothem method) | sq units |
Step-by-Step Examples
Example 1
Regular Hexagon
n=6, s=4.
- Interior angle=(6−2)×180/6=120°
- Apothem=4/(2tan30°)=4/1.155≈3.464
- Area=½×6×4×3.464≈41.57
✓ Area ≈ 41.57 sq units
Example 2
Pentagon
n=5, s=3.
- Interior angle=108°
- Area≈15.48 sq units
✓ Area ≈ 15.48 sq units
Example 3
Triangle
n=3, s=5.
- Interior angle=60°
- A=½×3×5×5/(2√3)≈5×5×√3/4≈10.83
✓ A ≈ 10.83 sq units
Real-World Applications
Architecture
Hexagonal tiling, octagonal windows, pentagonal structures.
Nature
Honeycombs are regular hexagons — optimal area-to-perimeter ratio.
Gaming
Hexagonal grids used in strategy games for fair movement in all directions.
Design
Regular polygons form the basis of many logos, tiles, and architectural plans.
Common Mistakes to Avoid
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n must be at least 3
A polygon needs at least 3 sides. n=1 and n=2 are not polygons.
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Side vs circumradius
This calculator uses side length. If given circumradius R, use s=2R×sin(π/n).
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Sum of interior angles
Total = (n−2)×180°. This equals each interior angle only for regular polygons (when divided by n).
Frequently Asked Questions
What is a regular polygon? ▾
A polygon with all sides equal (equilateral) and all angles equal (equiangular). Examples: equilateral triangle, square, regular pentagon, hexagon.
What is an apothem? ▾
The perpendicular distance from the center to the midpoint of any side. Area = ½×perimeter×apothem.
Why do bees use hexagons? ▾
Regular hexagons tile the plane perfectly, using the minimum perimeter (material) to enclose a given area. More efficient than squares or triangles.
What is the circumradius? ▾
The radius of the circumscribed circle (passing through all vertices). For regular n-gon with side s: R=s/(2sin(π/n)).
What happens as n→∞? ▾
The regular polygon approaches a circle. The area approaches πR² and the perimeter approaches 2πR.
Related Math Calculators
Formula Explorer connections
Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.