How to Draw Pentagon in a Circle

Shape with five sides

This commodity is about the geometric figure. For the headquarters of the United States Section of Defense, meet The Pentagon. For other uses, come across Pentagon (disambiguation).

Pentagon
An equilateral pentagon, i.e. a pentagon whose 5 sides all have the same length
Edges and vertices	5
Internal bending (degrees)	108° (if equiangular, including regular)

In geometry, a pentagon (from the Greek πέντε pente meaning v and γωνία gonia meaning angle ^[1]) is any 5-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.

Regular pentagons [edit]

Regular pentagon
A regular pentagon
Type	Regular polygon
Edges and vertices	5
Schläfli symbol	{5}
Coxeter–Dynkin diagrams
Symmetry grouping	Dihedral (D5), order 2×5
Internal bending (degrees)	108°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

A regular pentagon has Schläfli symbol {5} and interior angles of 108°.

A regular pentagon has 5 lines of reflectional symmetry, and rotational symmetry of lodge v (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length ${\displaystyle t,}$ its height ${\displaystyle H}$ (distance from 1 side to the opposite vertex), width ${\displaystyle Due west}$ (altitude between two farthest separated points, which equals the diagonal length ${\displaystyle D}$ ) and circumradius ${\displaystyle R}$ are given by:

${\displaystyle {\begin{aligned}H&={\frac {\sqrt {5+2{\sqrt {5}}}}{ii}}~t\approx one.539~t,\\W=D&={\frac {1+{\sqrt {five}}}{two}}~t\approx 1.618~t,\\W&={\sqrt {ii-{\frac {ii}{\sqrt {five}}}}}\cdot H\approx 1.051~H,\\R&={\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.8507~t,\\D&=R\ {\sqrt {\frac {5+{\sqrt {v}}}{2}}}=2R\cos xviii^{\circ }=2R\cos {\frac {\pi }{10}}\approx one.902~R.\end{aligned}}}$

The area of a convex regular pentagon with side length ${\displaystyle t}$ is given past

${\displaystyle {\begin{aligned}A&={\frac {t^{2}{\sqrt {25+10{\sqrt {5}}}}}{4}}={\frac {5t^{two}\tan 54^{\circ }}{4}}\\&={\frac {{\sqrt {v(v+2{\sqrt {five}})}}\;t^{2}}{4}}\approx 1.720~t^{2}.\end{aligned}}}$

If the circumradius ${\displaystyle R}$ of a regular pentagon is given, its edge length ${\displaystyle t}$ is constitute by the expression

${\displaystyle t=R\ {\sqrt {\frac {5-{\sqrt {5}}}{two}}}=2R\sin 36^{\circ }=2R\sin {\frac {\pi }{5}}\approx one.176~R,}$

and its area is

${\displaystyle A={\frac {5R^{2}}{4}}{\sqrt {\frac {5+{\sqrt {5}}}{two}}};}$

since the area of the circumscribed circle is ${\displaystyle \pi R^{2},}$ the regular pentagon fills approximately 0.7568 of its circumscribed circumvolve.

Derivation of the area formula [edit]

The area of any regular polygon is:

${\displaystyle A={\frac {1}{2}}Pr}$

where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula

${\displaystyle A={\frac {1}{2}}\cdot 5t\cdot {\frac {t\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{2}}={\frac {5t^{2}\tan {\mathord {\left({\frac {3\pi }{10}}\correct)}}}{iv}}}$

with side length t.

Inradius [edit]

Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by

${\displaystyle r={\frac {t}{2\tan {\mathord {\left({\frac {\pi }{5}}\right)}}}}={\frac {t}{two{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t.}$

Chords from the circumscribed circumvolve to the vertices [edit]

Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, Due east, if P is whatever signal on the circumcircle between points B and C, then PA + PD = Atomic number 82 + PC + PE.

Signal in plane [edit]

For an arbitrary indicate in the plane of a regular pentagon with circumradius ${\displaystyle R}$ , whose distances to the centroid of the regular pentagon and its five vertices are ${\displaystyle Fifty}$ and ${\displaystyle d_{i}}$ respectively, we accept ^[2]

${\displaystyle {\begin{aligned}\textstyle \sum _{i=1}^{5}d_{i}^{ii}&=5\left(R^{ii}+L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{4}&=5\left(\left(R^{2}+L^{two}\right)^{2}+2R^{2}L^{2}\correct),\\\textstyle \sum _{i=1}^{5}d_{i}^{six}&=5\left(\left(R^{2}+L^{2}\correct)^{iii}+6R^{ii}Fifty^{ii}\left(R^{2}+L^{ii}\right)\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{8}&=5\left(\left(R^{ii}+Fifty^{2}\right)^{4}+12R^{2}L^{2}\left(R^{2}+L^{ii}\right)^{2}+6R^{4}L^{4}\correct).\stop{aligned}}}$

If ${\displaystyle d_{i}}$ are the distances from the vertices of a regular pentagon to any betoken on its circumcircle, so ^[ii]

${\displaystyle 3\left(\textstyle \sum _{i=1}^{v}d_{i}^{2}\correct)^{ii}=10\textstyle \sum _{i=1}^{5}d_{i}^{four}.}$

Geometrical constructions [edit]

The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for amalgam a regular pentagon. Some are discussed below.

Richmond's method [edit]

One method to construct a regular pentagon in a given circle is described past Richmond^[3] and further discussed in Cromwell's Polyhedra.^[four]

The peak console shows the construction used in Richmond'southward method to create the side of the inscribed pentagon. The circle defining the pentagon has unit of measurement radius. Its centre is located at point C and a midpoint M is marked halfway along its radius. This signal is joined to the periphery vertically above the middle at point D. Bending CMD is bisected, and the bisector intersects the vertical axis at point Q. A horizontal line through Q intersects the circumvolve at signal P, and chord PD is the required side of the inscribed pentagon.

To decide the length of this side, the two right triangles DCM and QCM are depicted beneath the circle. Using Pythagoras' theorem and ii sides, the hypotenuse of the larger triangle is found equally ${\displaystyle \scriptstyle {\sqrt {5}}/2}$ . Side h of the smaller triangle so is found using the half-bending formula:

${\displaystyle \tan(\phi /2)={\frac {one-\cos(\phi )}{\sin(\phi )}}\ ,}$

where cosine and sine of ϕ are known from the larger triangle. The result is:

${\displaystyle h={\frac {{\sqrt {five}}-1}{4}}\ .}$

If DP is truly the side of a regular pentagon, ${\displaystyle m\angle \mathrm {CDP} =54^{\circ }}$ , so DP = two cos(54°), QD = DP cos(54°) = 2cos²(54°), and CQ = 1 − 2cos²(54°), which equals −cos(108°) by the cosine double angle formula. This is the cosine of 72°, which equals ${\displaystyle \left({\sqrt {5}}-1\correct)/four}$ as desired.

Carlyle circles [edit]

Method using Carlyle circles

The Carlyle circle was invented as a geometric method to discover the roots of a quadratic equation.^[5] This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:^[6]

1. Draw a circle in which to inscribe the pentagon and mark the center betoken O.
2. Depict a horizontal line through the center of the circle. Mark the left intersection with the circle every bit point B.
3. Construct a vertical line through the centre. Mark 1 intersection with the circle as point A.
4. Construct the point M as the midpoint of O and B.
5. Describe a circumvolve centered at Thou through the signal A. Mark its intersection with the horizontal line (inside the original circumvolve) as the point W and its intersection exterior the circle every bit the point Five.
6. Draw a circle of radius OA and eye W. It intersects the original circumvolve at two of the vertices of the pentagon.
7. Draw a circle of radius OA and center V. Information technology intersects the original circle at two of the vertices of the pentagon.
8. The fifth vertex is the rightmost intersection of the horizontal line with the original circle.

Steps vi–viii are equivalent to the following version, shown in the animation:

6a. Construct point F as the midpoint of O and Due west.

7a. Construct a vertical line through F. It intersects the original circumvolve at two of the vertices of the pentagon. The tertiary vertex is the rightmost intersection of the horizontal line with the original circumvolve.

8a. Construct the other two vertices using the compass and the length of the vertex found in pace 7a.

Euclid's method [edit]

Euclid's method for pentagon at a given circle, using of the golden triangle, animation 1 min 39 s

A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described past Euclid in his Elements circa 300 BC.^{[vii] [eight]}

Physical construction methods [edit]

Overhand knot of a paper strip

• A regular pentagon may be created from only a strip of paper past tying an overhand knot into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding 1 of the ends back over the pentagon will reveal a pentagram when backlit.
• Construct a regular hexagon on stiff paper or card. Crease along the three diameters betwixt opposite vertices. Cutting from 1 vertex to the center to make an equilateral triangular flap. Fix this flap underneath its neighbor to brand a pentagonal pyramid. The base of the pyramid is a regular pentagon.

Symmetry [edit]

Symmetries of a regular pentagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.

The regular pentagon has Dih₅ symmetry, order 10. Since 5 is a prime number there is i subgroup with dihedral symmetry: Dih₁, and ii cyclic group symmetries: Z₅, and Z₁.

These iv symmetries tin can exist seen in 4 singled-out symmetries on the pentagon. John Conway labels these by a letter and group order.^[9] Full symmetry of the regular course is r10 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as chiliad for their central gyration orders.

Each subgroup symmetry allows 1 or more than degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of liberty only can exist seen as directed edges.

Regular pentagram [edit]

A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides course the diagonals of a regular convex pentagon – in this arrangement the sides of the ii pentagons are in the golden ratio.

Equilateral pentagons [edit]

Equilateral pentagon built with four equal circles disposed in a chain.

An equilateral pentagon is a polygon with five sides of equal length. Nonetheless, its five internal angles tin have a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique upward to similarity, because it is equilateral and information technology is equiangular (its five angles are equal).

Circadian pentagons [edit]

A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an instance of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed equally 1 4th the square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon.^{[10] [11] [12]}

At that place exist circadian pentagons with rational sides and rational area; these are chosen Robbins pentagons. Information technology has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and information technology is conjectured that all the diagonals must be rational.^[thirteen]

General convex pentagons [edit]

For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.^{[14] : p.75, #1854}

Pentagons in tiling [edit]

A regular pentagon cannot appear in whatever tiling of regular polygons. First, to show a pentagon cannot course a regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 3 ane⁄three (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More than difficult is proving a pentagon cannot be in whatever edge-to-edge tiling made by regular polygons:

The maximum known packing density of a regular pentagon is approximately 0.921, accomplished by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner appear a proof that the double lattice packing of the regular pentagon (which they telephone call the "pentagonal water ice-ray" packing, and which they trace to the work of Chinese artisans in 1900) has the optimal density among all packings of regular pentagons in the plane.^[15] Every bit of 2020^[update], their proof has non even so been refereed and published.

There are no combinations of regular polygons with 4 or more meeting at a vertex that comprise a pentagon. For combinations with 3, if 3 polygons encounter at a vertex and i has an odd number of sides, the other 2 must be coinciding. The reason for this is that the polygons that affect the edges of the pentagon must alternate around the pentagon, which is impossible because of the pentagon'southward odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126°. To notice the number of sides this polygon has, the issue is 360 / (180 − 126) = 6 2⁄3 , which is not a whole number. Therefore, a pentagon cannot announced in whatsoever tiling made by regular polygons.

In that location are 15 classes of pentagons that can monohedrally tile the plane. None of the pentagons have whatsoever symmetry in general, although some have special cases with mirror symmetry.

15 monohedral pentagonal tiles

i	2	3	4	v
6	7	viii	9	10
eleven	12	xiii	xiv	15

Pentagons in polyhedra [edit]

Ih	Th	Td	O	I	D5d
Dodecahedron	Pyritohedron	Tetartoid	Pentagonal icositetrahedron	Pentagonal hexecontahedron	Truncated trapezohedron

Pentagons in nature [edit]

Plants [edit]

• 

  Pentagonal cantankerous-section of okra.

• 

  Starfruit is another fruit with fivefold symmetry.

Animals [edit]

• 

  Another example of echinoderm, a sea urchin endoskeleton.

• 

  An illustration of brittle stars, also echinoderms with a pentagonal shape.

Minerals [edit]

• 

  A pyritohedral crystal of pyrite. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.

Other examples [edit]

• 

• 

See besides [edit]

• Associahedron; A pentagon is an lodge-4 associahedron
• Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal faces
• Golden ratio
• List of geometric shapes
• Pentagonal numbers
• Pentagram
• Pentagram map
• Pentastar, the Chrysler logo
• Pythagoras' theorem#Similar figures on the three sides
• Trigonometric constants for a pentagon

In-line notes and references [edit]

1. ^ "pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 Baronial 2014.
2. ^ ^{a b} Meskhishvili, Mamuka (2020). "Circadian Averages of Regular Polygons and Ideal Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340.
3. ^ Herbert W Richmond (1893). "Pentagon".
4. ^ Peter R. Cromwell (22 July 1999). Polyhedra. p. 63. ISBN0-521-66405-5.
5. ^ Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 329. ISBN1-58488-347-2.
6. ^ DeTemple, Duane W. (February 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF). The American Mathematical Monthly. 98 (2): 97–108. doi:x.2307/2323939. JSTOR 2323939. Archived from the original (PDF) on 2015-12-21.
7. ^ George Edward Martin (1998). Geometric constructions. Springer. p. 6. ISBN0-387-98276-0.
8. ^ Fitzpatrick, Richard (2008). Euklid's Elements of Geometry, Volume 4, Proposition 11 (PDF). Translated by Richard Fitzpatrick. p. 119. ISBN978-0-6151-7984-1.
9. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter twenty, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
10. ^ Weisstein, Eric West. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. [1]
11. ^ Robbins, D. P. (1994). "Areas of Polygons Inscribed in a Circle". Discrete and Computational Geometry. 12 (ii): 223–236. doi:10.1007/bf02574377.
12. ^ Robbins, D. P. (1995). "Areas of Polygons Inscribed in a Circle". The American Mathematical Monthly. 102 (half-dozen): 523–530. doi:10.2307/2974766. JSTOR 2974766.
13. ^ *Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and surface area", Periodical of Number Theory, 128 (ane): 17–48, doi:10.1016/j.jnt.2007.05.005, MR 2382768 .
14. ^ Inequalities proposed in "Crux Mathematicorum", [two].
15. ^ Hales, Thomas; Kusner, Wöden (September 2016), Packings of regular pentagons in the plane, arXiv:1602.07220

External links [edit]

Look upwardly pentagon in Wiktionary, the gratuitous dictionary.

Wikimedia Commons has media related to Pentagons.

• Weisstein, Eric W. "Pentagon". MathWorld.
• Blithe sit-in constructing an inscribed pentagon with compass and straightedge.
• How to construct a regular pentagon with simply a compass and straightedge.
• How to fold a regular pentagon using only a strip of paper
• Definition and properties of the pentagon, with interactive animation
• Renaissance artists' approximate constructions of regular pentagons
• Pentagon. How to calculate diverse dimensions of regular pentagons.

v t eastward Fundamental convex regular and uniform polytopes in dimensions ii–10
Family	A n	B north	I 2(p) / D n	E 6 / E 7 / Eastward 8 / F iv / Thou 2	Hnorthward
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-jail cell	120-prison cell • 600-prison cell
Uniform v-polytope	5-simplex	5-orthoplex • 5-cube	five-demicube
Compatible 6-polytope	6-simplex	6-orthoplex • six-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	seven-demicube	one32 • two31 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	viii-demicube	142 • 241 • 421
Compatible 9-polytope	ix-simplex	nine-orthoplex • 9-cube	9-demicube
Compatible 10-polytope	10-simplex	x-orthoplex • 10-cube	10-demicube
Uniform n-polytope	north-simplex	north-orthoplex • n-cube	due north-demicube	ik2 • 2k1 • grand21	north-pentagonal polytope
Topics: Polytope families • Regular polytope • Listing of regular polytopes and compounds

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Source: https://en.wikipedia.org/wiki/Pentagon

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