How to compute area and perimeter of a regular polygon: geometric figure with n equal sides and n equal interior angles. Area formula: A = (1/2) * P * a = (n * s²) / (4 * tan(pi/n)), where P=n*s perimeter, s side, a apothem (center-to-side distance), n number of sides. Apothem formula: a = s / (2 * tan(pi/n)). Circumradius (circumscribed circle): R = s / (2 * sin(pi/n)). Side from R: s = 2 * R * sin(pi/n). Essential tool for school geometry (high schools, middle schools), architecture (hexagonal beehives, octagonal Mole Antonelliana Turin), CAD engineering (SolidWorks AutoCAD), tessellation paving tiles (Florim Marazzi USA), optics lens diaphragms cameras, industrial design (hex M nuts wrenches, screws). Equilateral triangle n=3: A=s²*sqrt(3)/4, hexagon n=6: A=s²*3*sqrt(3)/2, perimeter most efficient material form.

Regular polygons formulas

Side, apothem, circumradius relations: for n-sided regular polygon, derivable from each other: a = s / (2*tan(pi/n)), R = s / (2*sin(pi/n)), s = 2*R*sin(pi/n) = 2*a*tan(pi/n). Example hexagon n=6, s=10: a=8.66, R=10. Hexagon unique characteristic: R=s (circumradius equals side).

Area: A = (1/2) * P * a, with P=n*s perimeter. Equivalent: A = (n * s²) / (4 * tan(pi/n)) = (1/2) * n * R² * sin(2*pi/n) = n * a² * tan(pi/n). Example hexagon s=10: A = 259.8. Example dodecagon n=12 s=5: A = 279.9.

Angles: interior angle = (n-2)*180/n degrees (interior sum = (n-2)*180). Exterior angle = 360/n. Central angle = 360/n (equal to exterior). Example pentagon n=5: interior=108°, exterior=72°. Hexagon n=6: interior=120° (beehive efficiency). Octagon n=8: interior=135°.

Diagonals: number of diagonals D = n*(n-3)/2. Triangle (n=3): D=0. Square n=4: D=2. Pentagon n=5: D=5. Hexagon n=6: D=9. Triangulation: n-2 triangles (useful for Gauss surveyor's formula area).

Compass and straightedge construction: Gauss 1796 (age 19) proved n-gon regular constructible if and only if n=2^k * p1 * p2 * ... * pm where p_i Fermat primes (3, 5, 17, 257, 65537). So triangle, square, pentagon, hexagon, octagon, decagon YES constructible. Heptagon n=7 NO (not Fermat). 17-gon Gauss-constructible Disquisitiones Arithmeticae 1801. 257-gon and 65537-gon theoretically constructible.

History and context

Study of regular polygons dates to Euclid (Greece 300 BC) "Elements" Book IV systematic treatment compass-and-straightedge constructions. Pythagoras (Greece 570-495 BC) discovered regular dodecahedron and connection with golden ratio phi=(1+sqrt(5))/2. Archimedes (287-212 BC) Syracuse pi calculation approximating circle with regular polygons up to 96 sides: 3.1408 < pi < 3.1429. Carl Friedrich Gauss (Germany 1777-1855) demonstrated 17-gon construction 1796 at age 19, "Disquisitiones Arithmeticae" 1801. Felix Klein (1849-1925) and David Hilbert (1862-1943) Gottingen modern geometry. USA: MIT Cambridge MA, Princeton NJ, Caltech Pasadena geometry research. Famous structures: Pentagon US Department of Defense Arlington Virginia (Bergmann-Lloyd 1943) iconic 5-sided building, perimeter 1405 m. Stop sign American 8-sided octagon (Marshall Michigan 1922 standardized MUTCD). Hexagonal beehives: Charles Darwin "On the Origin of Species" 1859 praised "absolutely perfect" engineering. Graphene hexagonal C-C 1.42 Å bond, Geim-Novoselov Nobel Physics 2010 Manchester UK. USA architecture: Buckminster Fuller (USA 1895-1983) geodesic domes 1947 NYC, Frank Lloyd Wright Fallingwater PA. Industrial: Boeing Seattle, Lockheed Martin Bethesda hexagonal honeycomb composite aerospace structures. Carbon nanotubes hexagonal Iijima 1991 NEC Tokyo. CAD industry: AutoCAD Autodesk San Rafael CA 1982, SolidWorks Dassault Systemes 1995.

Mathematics: AMS American Mathematical Society Providence RI, MAA Mathematical Association America. Crystallography: ASA American Society Astronomy hexagonal close-packed metals (Mg, Zn, Ti).

Regular polygons properties table

• Equilateral triangle (n=3): interior angle 60°, A=s²*sqrt(3)/4, a=s*sqrt(3)/6, R=s*sqrt(3)/3
• Square (n=4): interior angle 90°, A=s², a=s/2, R=s*sqrt(2)/2
• Pentagon (n=5): interior angle 108°, A=s²*sqrt(25+10*sqrt(5))/4, R=s*phi/sqrt(2)
• Hexagon (n=6): interior angle 120°, A=s²*3*sqrt(3)/2, a=s*sqrt(3)/2, R=s
• Heptagon (n=7): interior angle 128.57°, NO compass-straightedge construction
• Octagon (n=8): interior angle 135°, A=2*s²*(1+sqrt(2)), a=s*(1+sqrt(2))/2
• Nonagon (n=9): interior angle 140°, NO compass-straightedge construction
• Decagon (n=10): interior angle 144°, A=s²*5*sqrt(5+2*sqrt(5))/2
• Dodecagon (n=12): interior angle 150°, constructible compass-straightedge
• 17-gon: interior angle 158.82°, Gauss 1796 construction proven
• Icosagon (n=20): interior angle 162°, constructible
• n=infinity (circle): A=pi*r², P=2*pi*r, regular polygon limit
• Triangle D=0, square D=2, pentagon D=5, hexagon D=9 diagonals
• Golden ratio phi=(1+sqrt(5))/2 ≈ 1.618: pentagon diagonal/side
• Hexagon special: R=s circumradius equals side
• Triangulation: n-2 triangles (useful surveyor's formula area)

Worked numerical examples

Example 1 - regular hexagon beehive cell. n=6, s=10 mm (typical cell). a=s/(2*tan(30°))=8.660 mm. R=s/(2*sin(30°))=10 mm (R=s hexagon characteristic). P=60 mm. A=259.8 mm². Hexagon vs inscribed circle same a: A_circle=pi*8.660²=235.6 mm² (hexagon larger area). Hexagon vs square side 10: A=100 mm² (hexagon 2.6x area). Hexagon maximum area per perimeter tessellable (bees, graphene, hydrogen graphite). Darwin praised "absolutely perfect" engineering 1859.

Example 2 - regular pentagon given circumradius. n=5, R=10 cm. s=2*R*sin(pi/5)=11.756 cm. a=R*cos(pi/5)=8.090 cm. P=58.78 cm. A=237.8 cm². Golden ratio: diagonal/side = phi=(1+sqrt(5))/2=1.618. Pentagon US Department of Defense Arlington Virginia 1943 (Bergmann-Lloyd architects), 5 sides x 281 m each, perimeter 1405 m, area 117 km² including parks.

Example 3 - regular octagon given apothem. n=8, a=5 m. s=2*a*tan(pi/8)=4.142 m. R=a/cos(pi/8)=5.412 m. P=33.14 m. A=82.84 m². American Stop sign (Marshall Michigan 1922 standard MUTCD): n=8, s=12.5"=31.75 cm, a=15"=38.1 cm, A=950 in²=6126 cm². Sun Moon stones Stonehenge outer circle.

Example 4 - equilateral triangle. n=3, s=10. a=2.887. R=5.774. P=30. A=43.30 (formula direct A=s²*sqrt(3)/4). GPS triangulation: 3 satellites minimum 2D geolocation, 4 for 3D. Buckminster Fuller geodesic domes 1947 USA Industrial Design.

Example 5 - regular dodecagon 12 sides. n=12, s=5 cm. a=9.330 cm. R=9.659 cm. P=60 cm. A=279.9 cm². Interior angle 150°. Ratio area/inscribed circle=A/(pi*a²)=1.024 (98% inscribed circle area: near circle limit). 12-fold rotational symmetry, ancient temples Maya/Aztec dodecagonal calendars.

Limits and considerations

1. Circle approximation: for n>>1 area->pi*R². Example n=100: A/A_circle=99.97%. n=1000: 99.9997%.
2. Compass-straightedge construction: only Fermat primes (3, 5, 17, 257, 65537) products with powers 2. Heptagon n=7 NO.
3. Star polygons (pentagram): not regular polygons, sides cross.
4. Regular plane tessellation: only triangle, square, hexagon. No pentagon (gaps left).
5. 3D tessellation: cube (square), parallelepiped. No regular tetrahedron (requires octahedron).
6. Historical pi approximation: Archimedes 96-gon 3.1408 < pi < 3.1429.
7. Golden ratio: pentagon diagonal/side = phi=(1+sqrt(5))/2.
8. Hexagon unique R=s characteristic, favors beehives, graphene, palettes.
9. Triangulation: every n-gon triangulates into n-2 triangles (Gauss surveyor's formula).
10. Apothem vs inscribed radius: a=apothem=inscribed circle radius, R=circumradius=circumscribed circle.

Common errors

1. Confusing apothem a (center-side) with circumradius R (center-vertex).
2. Calculating interior angle with (n-2)*180 instead (n-2)*180/n (sum vs single).
3. Demanding compass-straightedge construction for n=7, 9, 11, 13 (not Fermat primes).
4. Calculating pentagon area with square approximation (27% error).
5. Confusing apothem with circumradius: hexagon equal, but other polygons R>a.
6. Neglecting units: mm² vs cm² vs m², 10000 factor error.
7. Calculating diagonals with n*(n-1)/2 (total sides) instead n*(n-3)/2 (non-adjacent diagonals).
8. Demanding regular pentagon tessellate plane: NO, leaves gaps, only n=3,4,6 tessellate.
9. Calculating pi without adequate n: n=10 error 5%, n=100 error 0.03%.
10. Ignoring golden ratio pentagon: diagonal=phi*side classical.

Numbered applications

1. Architecture: Pentagon US DoD, Buckminster Fuller geodesic domes, Frank Lloyd Wright.
2. Industrial design: hex M nuts ISO 4014/ANSI, octagonal Stop signs MUTCD American.
3. Ceramics and tessellations: tile manufacturers Florim Marazzi USA distribution.
4. Beekeeping biomimicry: hexagonal beehives, biomimicry study Janine Benyus 1997.
5. Materials science: hexagonal graphene C-C, hexagonal close-packed metals (Mg, Zn, Ti).
6. Crystallography: 6-fold hexagonal symmetry, P6 space group crystals.
7. Optics: hexagonal aperture iris cameras 6-9 blade, F-stop f/1.4 to f/22.
8. Game design: hex-grid wargames, Catan board game hexagonal.
9. Math history: Gauss 17-gon 1796, Disquisitiones Arithmeticae 1801.
10. Educational: high school geometry curriculum, MAA AMS American Mathematical.

For support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary plane geometry. For professional applications consult Euclid "Elements" Book IV, Coxeter "Regular Polytopes" 1973 Dover, Gauss "Disquisitiones Arithmeticae" 1801, AutoCAD Autodesk manual, MUTCD Manual Uniform Traffic Control Devices USA. ANSI/ASME B18.2.1 hex screws and nuts. AMS Mathematical Society Providence RI. ISO 4014 hexagonal nuts mechanical screws standard.

Updated July 2026