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Regular Polygon

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RegularPolygons

A regular polygon is an n-sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral). Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.

The terms equilateral triangle and square refer to the regular 3- and 4-polygons, respectively. The words for polygons with n>=5 sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, although the terms generally refer to regular polygons in the absence of specific wording.

A regular n-gon is implemented in the Wolfram Language as RegularPolygon[n], or more generally as RegularPolygon[r, n], RegularPolygon[{x, y}, rspec, n], etc.

The sum of perpendiculars from any point to the sides of a regular polygon of n sides is n times the apothem.

PolygonInCircumscribe

Let a be the side length, r be the inradius, and R the circumradius of a regular polygon. Then

a=2rtan(pi/n)
(1)
=2Rsin(pi/n)
(2)
r=1/2acot(pi/n)
(3)
=Rcos(pi/n)
(4)
R=1/2acsc(pi/n)
(5)
=rsec(pi/n)
(6)
A=1/4na^2cot(pi/n)
(7)
=nr^2tan(pi/n)
(8)
=1/2nR^2sin((2pi)/n).
(9)

The area moments of inertia about axes along an inradius and a circumradius of a regular n-gon are given by

I_r=1/(24)A_n(6r_n^2-a^2)
(10)
=(a^4)/(192)n[cos((2pi)/n)+2]cos(pi/n)csc^2(pi/n)
(11)
I_R=1/(48)A_n(12R_n^2+a^2)
(12)
=(a^4)/(192)ncot(pi/n)[3cos^2(pi/n)+1]
(13)

(Roark 1954, p. 70).

If the number of sides is doubled, then

a_(2n)=sqrt(2R^2-Rsqrt(4R^2-a_n^2))
(14)
A_(2n)=(4rA_n)/(2r+sqrt(4r^2+a_n^2)).
(15)

The area of the first few regular n-gon with unit edge lengths are

A_3=1/4sqrt(3)
(16)
A_4=1
(17)
A_5=1/4sqrt(5(5+2sqrt(5)))
(18)
A_6=3/2sqrt(3)
(19)
A_7=(4096x^6-62720x^4+115248x^2-16807)_6
(20)
A_8=2(1+sqrt(2))
(21)
A_9=(4096x^6-186624x^4+1154736x^2-177147)_6
(22)
A_(10)=5/2sqrt(5+2sqrt(5)).
(23)

The algebraic degrees of these for n=3, 4, ... are 2, 1, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 8, 4, 16, 6, 18, 4, ... (OEIS A089929).

RegularPolygonAreas

The plot above shows how the areas of the regular n-gons with unit inradius (blue) and unit circumradius (red) approach that of a unit disk (i.e., pi).

If p_k and P_k are the perimeters of the regular polygons inscribed in and circumscribed around a given circle and a_k and A_k their areas, then

P_(2n)=(2p_nP_n)/(p_n+P_n)
(24)
p_(2n)=sqrt(p_nP_(2n)),
(25)

and

a_(2n)=sqrt(a_nA_n)
(26)
A_(2n)=(2a_(2n)A_n)/(a_(2n)+A_n)
(27)

(Beyer 1987, p. 125).

The sum of interior angles in any n-gon is given by (n-2)pi radians, or 2(n-2)×90 degrees (Zwillinger 1995, p. 270).

The following table gives parameters for the first few regular polygons of unit edge length s=1, where alpha is the interior (vertex) angle, beta is the exterior angle, r is the inradius, R is the circumradius, and A is the area (Williams 1979, p. 33).

polygon{n}alphabetarRA
equilateral triangle{3}1/3pi=60 degrees2/3pi=120 degrees1/6sqrt(3)1/3sqrt(3)1/4sqrt(3)
square{4}1/2pi=90 degrees1/2pi=90 degrees1/21/2sqrt(2)1
pentagon{5}3/5pi=108 degrees2/5pi=72 degrees1/(10)sqrt(25+10sqrt(5))1/(10)sqrt(50+10sqrt(5))1/4sqrt(25+10sqrt(5))
hexagon{6}2/3pi=120 degrees1/3pi=60 degrees1/2sqrt(3)13/2sqrt(3)
heptagon{7}5/7pi=(900)/7 degrees2/7pi=(360)/7 degrees1/2cot(1/7pi)1/2csc(1/7pi)7/4cot(1/7pi)
octagon{8}3/4pi=135 degrees1/4pi=45 degrees1/2(1+sqrt(2))1/2sqrt(4+2sqrt(2))2(1+sqrt(2))
nonagon{9}7/9pi=140 degrees2/9pi=40 degrees1/2cot(1/9pi)1/2csc(1/9pi)9/4cot(1/9pi)
decagon{10}4/5pi=144 degrees1/5pi=36 degrees1/2sqrt(5+2sqrt(5))1/2(1+sqrt(5))5/2sqrt(5+2sqrt(5))
hendecagon{11}9/(11)pi=(1620)/(11) degrees2/(11)pi=(360)/(11) degrees1/2cot(1/(11)pi)1/2csc(1/(11)pi)(11)/4cot(1/(11)pi)
dodecagon{12}5/6pi=150 degrees1/6=30 degrees1/2(2+sqrt(3))1/2(sqrt(2)+sqrt(6))3(2+sqrt(3))
tridecagon{13}(11)/(13)pi=(1980)/(13) degrees2/(13)pi=(360)/(13) degrees1/2cot(1/(13)pi)1/2csc(1/(13)pi)(13)/4cot(1/(13)pi)
tetradecagon{14}6/7pi=(1080)/7 degrees1/7pi=(180)/7 degrees1/2cot(1/(14)pi)1/2csc(1/(14)pi)7/2cot(1/(14)pi)

Only some of the regular polygons can be built by geometric construction using a compass and straightedge. The numbers of sides for which regular polygons are constructible are those having central angles corresponding to so-called trigonometry angles.

RegularPolygonFunctions

It is possible to construct relatively simple two-dimensional functions P_n(x,y) that have the symmetry of a regular n-gon (i.e., whose level curves are regular n-gons). Examples, illustrated above, include

P_3(x,y)=max(y-xsqrt(3),y+xsqrt(3),-2y)
(28)
P_4(x,y)=|x|+|y|
(29)
P_6(x,y)=2|x|+|x-ysqrt(3)|+|x+ysqrt(3)|
(30)
P_8(x,y)=2(|x|+|y|)+sqrt(2)(|x-y|+|x+y|).
(31)

See also

257-gon, 65537-gon, Apeirogon, Bill Picture, Chaos Game, Constructible Polygon, de Moivre Number, Equilateral Triangle, Heptadecagon, Hexagon, Hexagram, Octagon, Pentagon, Pentagram, Polygon, Polygon Circumscribing, Polygon Inscribing, Regular Polygon Division by Diagonals, Square, Star Polygon, Trigonometry Angles Explore this topic in the MathWorld classroom

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Bishop, W. "How to Construct a Regular Polygon." Amer. Math. Monthly 85, 186-188, 1978.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 140 and 197-202, 1996.Courant, R. and Robbins, H. "Regular Polygons." §3.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 122-125, 1996.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.DeTemple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991.Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352-386, 1955.Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, p. 207, 1977.Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. Translated by A. A. Clarke. New Haven, CT: Yale University Press, 1965.Harris, J. W. and Stocker, H. "Regular n-gons (Polygons)." §3.7 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 86-89, 1998.Math Forum. "Naming Polygons and Polyhedra." https://mathforum.org/dr.math/faq/faq.polygon.names.html.Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 238, 1997.Richmond, H. W. "A Construction for a Regular Polygon of Seventeen Sides." Quart. J. Pure Appl. Math. 26, 206-207, 1893.Roark, R. J. Formulas for Stress and Strain, 3rd ed. New York: McGraw-Hill, 1954.Sloane, N. J. A. Sequences A003401/M0505, A004729, and A089929 in "The On-Line Encyclopedia of Integer Sequences."Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 350, 1994.Tietze, H. Ch. 9 in Famous Problems of Mathematics. New York: Graylock Press, 1965.Wantzel, M. L. "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas." J. Math. pures appliq. 1, 366-372, 1836.Williams, R. "Polygons." §2-1 in The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, pp. 31-33, 1979.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.

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Regular Polygon

Cite this as:

Weisstein, Eric W. "Regular Polygon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularPolygon.html

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