Regular Polygon Area Calculator 2025: Complete Guide (1000+ Words)
A regular polygon has equal sides and angles. Our 2025 Regular Polygon Area Calculator computes area, perimeter, apothem, and circumradius from number of sides and either side length or circumradius with full step-by-step solutions and interactive graphics.
What is a Regular Polygon?
All sides equal, all interior angles equal. Examples: equilateral triangle, square, regular pentagon, hexagon.
- Sides (n): n ≥ 3
- Side Length (s): Length of one side
- Circumradius (R): Distance from center to vertex
- Apothem (a): Distance from center to midpoint of side
Core Formulas
Area (from side): A = (1/4) n s² cot(π/n)
Area (from radius): A = (1/2) n R² sin(2π/n)
Perimeter: P = n s
Apothem: a = R cos(π/n)
Side from radius: s = 2 R sin(π/n)
Area (from radius): A = (1/2) n R² sin(2π/n)
Perimeter: P = n s
Apothem: a = R cos(π/n)
Side from radius: s = 2 R sin(π/n)
Example: n=6, s=8 cm
A = (1/4)×6×64×cot(30°) approximately 166.277 cm²
P = 6×8 = 48 cm
R = 8 / (2×sin(30°)) = 8 cm
a = 8×cos(30°) approximately 6.928 cm
P = 6×8 = 48 cm
R = 8 / (2×sin(30°)) = 8 cm
a = 8×cos(30°) approximately 6.928 cm
Applications
- Architecture: Floor tiles, windows
- Engineering: Gears, bolts, nuts
- Design: Logos, patterns
- Nature: Honeycombs, crystals
Step-by-Step: n=5, R=10 m
A = (1/2)×5×100×sin(72°) approximately 475.528 m²
s = 2×10×sin(36°) approximately 11.756 m
Common Mistakes
- Using inradius instead of circumradius
- Forgetting to convert degrees to radians in trig functions
- Using n=4 for triangle
Advanced: Interior Angle
Interior angle = ((n-2)×180°)/n
Related Tools
FAQs
Can n be decimal? No, n must be integer ≥ 3.
Is circle a polygon? No, infinite sides.
Conclusion
Regular polygons are foundational in geometry. Our 2025 calculator delivers instant precision with crystal-clear steps and interactive graphics. Master polygonal calculations today!
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