Regular pentagon

A regular pentagon

Type Regular polygon

Edges andvertices

5

Schläfli symbol {5}

Coxeter diagram

Symmetry group Dihedral (D5), order 2×5

Internal angle(degrees)

108°

Dual polygon self

Properties convex, cyclic, equilateral,isogonal, isotoxal

PentagonFrom Wikipedia, the free encyclopedia

In geometry, a pentagon (from pente, which is Greek for thenumber 5) is any five-sided polygon. A pentagon may besimple or self-intersecting. The sum of the internal angles ina simple pentagon is 540°. A pentagram is an example of aself-intersecting pentagon.

Contents

1 Regular pentagons1.1 Derivation of the area formula1.2 Derivation of the diagonal length formula1.3 Chords from the circumscribing circle tothe vertices

2 Construction of a regular pentagon2.1 Euclid's methods2.2 Richmond's method2.3 Verification2.4 Alternative method2.5 Carlyle circles2.6 Direct method2.7 Simple methods

3 Cyclic pentagons4 Graphs5 Pentagons in nature

5.1 Plants5.2 Animals

6 Pentagons in tiling7 See also8 In-line notes and references9 External links

Regular pentagons

In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has fivelines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). ItsSchläfli symbol is {5}. The diagonals of a regular pentagon are in golden ratio to its sides.

The area of a regular convex pentagon with side length t is given by

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

When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression

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Derivation of the area formula

The area of any regular polygon is:

where P is the perimeter of the polygon, a is the apothem. One can then substitute the respective values for Pand a, which makes the formula:

with t as the given side length. Then we can then rearrange the formula as:

and then, we combine the two terms to get the final formula, which is:

Derivation of the diagonal length formula

The diagonals of a regular pentagon (hereby represented by D) can be calculated based upon the golden ratio f�and the known side T (see discussion of the pentagon in Golden ratio):

Accordingly:

Chords from the circumscribing circle to the vertices

If a regular pentagon with successive vertices A, B, C, D, E is inscribed in a circle, and if P is any point on thatcircle between points B and C, then PA + PD = PB + PC + PE.

Construction of a regular pentagon

A variety of methods are known for constructing a regular pentagon. Some are discussed below.

Euclid's methods

A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circleor constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.[1]

Richmond's method

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One method to construct a regular pentagon in a given circle is described by Richmond[2] and further discussedin Cromwell's "Polyhedra."[3]

Verification

The top panel describes the construction used in the animation above to createthe side of the inscribed pentagon. The circle defining the pentagon has unitradius. Its center is located at point C and a midpoint M is marked halfway alongits radius. This point is joined to the periphery vertically above the center at pointD. Angle CMD is bisected, and the bisector intersects the vertical axis at point Q.A horizontal line through Q intersects the circle at point P, and chord PD is therequired side of the inscribed pentagon.

To determine the length of this side, the two right triangles DCM and QCM aredepicted below the circle. Using Pythagoras' theorem and two sides, thehypotenuse of the larger triangle is found as . Side h of the smaller trianglethen is found using the half-angle formula:

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

With this side known, attention turns to the lower diagram to find the side s of theregular pentagon. First, side a of the right-hand triangle is found usingPythagoras' theorem again:

Then s is found using Pythagoras' theorem and the left-hand triangle as:

The side s is therefore:

a well established result.[4] Consequently, this construction of the pentagon is valid.

Alternative method

An alternative method is this:

Draw a circle in which to inscribe the pentagon and mark the center point O. (This is the green circle inthe diagram to the right).

1.

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Constructing a pentagon

Method using Carlyle circles

Choose a point A on the circle that will serve as one vertex of thepentagon. Draw a line through O and A.

2.

Construct a line perpendicular to the line OA passing through O.Mark its intersection with one side of the circle as the point B.

3.

Construct the point C as the midpoint of the line OB.4.Draw a circle centered at C through the point A. Mark itsintersection with the line OB (inside the original circle) as thepoint D.

5.

Draw a circle centered at A through the point D. Mark itsintersections with the original (green) circle as the points E and F.

6.

Draw a circle centered at E through the point A. Mark its otherintersection with the original circle as the point G.

7.

Draw a circle centered at F through the point A. Mark its otherintersection with the original circle as the point H.

8.

Construct the regular pentagon AEGHF.9.

Animation that is almost the same as thisalternative method

Carlyle circles

The Carlyle circle was invented as a geometric method to find the rootsof a quadratic equation.[5] This methodology leads to a procedure forconstructing a regular pentagon. The steps are as follows:[6]

Draw a circle in which to inscribe the pentagon and mark thecenter point O.

1.

Draw a horizontal line through the center of the circle. Mark oneintersection with the circle as point B.

2.

Construct a vertical line through the center. Mark one intersectionwith the circle as point A.

3.

Construct the point M as the midpoint of O and B.4.Draw a circle centered at M through the point A. Mark itsintersection with the horizontal line (inside the original circle) asthe point W and its intersection outside the circle as the point V.

5.

Draw a circle of radius OA and center W. It intersects the originalcircle at two of the vertices of the pentagon.

6.

Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of thepentagon.

7.

The fifth vertex is the intersection of the horizontal axis with the original circle.8.

Direct method

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Overhand knotof a paper strip

A direct method using degrees follows:

Draw a circle and choose a point to be the pentagon's (e.g. top center)1.Choose a point A on the circle that will serve as one vertex of the pentagon. Draw a line through O andA.

2.

Draw a guideline through it and the circle's center3.Draw lines at 54° (from the guideline) intersecting the pentagon's point4.Where those intersect the circle, draw lines at 18° (from parallels to the guideline)5.Join where they intersect the circle6.

After forming a regular convex pentagon, if one joins the non-adjacent corners (drawing the diagonals of thepentagon), one obtains a pentagram, with a smaller regular pentagon in the center. Or if one extends the sidesuntil the non-adjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on theaccuracy of the protractor used to measure the angles.

Simple methods

A regular pentagon may be created from just a strip of paper by tying an overhandknot into the strip and carefully flattening the knot by pulling the ends of the paperstrip. Folding one of the ends back over the pentagon will reveal a pentagram whenbacklit.Construct a regular hexagon on stiff paper or card. Crease along the three diametersbetween opposite vertices. Cut from one vertex to the center to make an equilateraltriangular flap. Fix this flap underneath its neighbor to make a pentagonal pyramid.The base of the pyramid is a regular pentagon.

Cyclic pentagons

A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regularpentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can beexpressed as one fourth the square root of one of the roots of a septic equation whose coefficients are functionsof the sides of the pentagon.[7][8][9]

There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons. In aRobbins pentagon, either all diagonals are rational or all are irrational, and it is conjectured that all thediagonals must be rational.[10]

Graphs

The K5 complete graph is often drawn as a regular pentagon with all 10 edges connected. This graph alsorepresents an orthographic projection of the 5 vertices and 10 edges of the 5-cell. The rectified 5-cell, withvertices at the mid-edges of the 5-cell is projected inside a pentagon.

5-cell (4D) Rectified 5-cell (4D)

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The best known packing of equal-sizedregular pentagons on a plane is adouble lattice structure which covers

Pentagons in nature

Plants

Pentagonal cross-section of okra.

Morning glories, likemany other flowers,have a pentagonalshape.

The gynoecium of anapple contains fivecarpels, arranged in afive-pointed star

Starfruit is another fruitwith fivefoldsymmetry.

Animals

A sea star. Manyechinoderms havefivefold radialsymmetry.

An illustration of brittlestars, also echinodermswith a pentagonalshape.

Pentagons in tiling

Main article: Pentagon tiling

A pentagon cannot appear in any tiling made by regular polygons. Toprove a pentagon cannot form a regular tiling (one in which all faces arecongruent), observe that 360 / 108 = 31/�3, which is not a whole number.More difficult is proving a pentagon cannot be in any edge-to-edgetiling made by regular polygons:

There are no combinations of regular polygons with 4 or more meetingat a vertex that contain a pentagon. For combinations with 3, if 3polygons meet at a vertex and one has an odd number of sides, the other2 must be congruent. The reason for this is that the polygons that touchthe edges of the pentagon must alternate around the pentagon, which isimpossible because of the pentagon's odd number of sides. For thepentagon, this results in a polygon whose angles are all(360 -� 108) / 2 = 126°. To find the number of sides this polygon has, the

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92.131% of the plane.result is 360 / (180 -� 126) = 62/�3, which is not a whole number.Therefore, a pentagon cannot appear in any tiling made by regularpolygons.

See also

Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal facesTrigonometric constants for a pentagonPentagonal numbersAssociahedron; A pentagon is an order-4 associahedronPentagramPentastar, the Chrysler logoPentagram mapGolden ratioPythagoras' theoremList of geometric shapes

In-line notes and references^ George Edward Martin (1998). Geometric constructions (http://books.google.com/books?id=ABLtD3IE_RQC&pg=PA6) . Springer. p. 6. ISBN 0-387-98276-0. http://books.google.com/books?id=ABLtD3IE_RQC&pg=PA6.

1.

^ Herbert W Richmond (1893). "Pentagon" (http://mathworld.wolfram.com/Pentagon.html) .http://mathworld.wolfram.com/Pentagon.html.

2.

^ Peter R. Cromwell. Polyhedra (http://books.google.com/books?id=OJowej1QWpoC&pg=PA63) . p. 63.ISBN 0-521-66405-5. http://books.google.com/books?id=OJowej1QWpoC&pg=PA63.

3.

^ This result agrees with Herbert Edwin Hawkes, William Arthur Luby, Frank Charles Touton (1920). "Exercise175" (http://books.google.com/books?id=eOdHAAAAIAAJ&pg=PA302) . Plane geometry. Ginn & Co.. p. 302.http://books.google.com/books?id=eOdHAAAAIAAJ&pg=PA302.

4.

^ Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (http://books.google.com/books?id=Zg1_QZsylysC&pg=PA329) (2nd ed.). CRC Press. p. 329. ISBN 1-58488-347-2.http://books.google.com/books?id=Zg1_QZsylysC&pg=PA329.

5.

^ Duane W DeTemple (1991). "Carlyle Circles and the Lemoine Simplicity of Polygon Constructions"(http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf) . The American Mathematical Monthly 98(2): 97–108. http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf. JSTOR link(http://www.jstor.org/stable/2323939)

6.

^ Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. [1](http://mathworld.wolfram.com/CyclicPentagon.html)

7.

^ Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223-236, 1994.8.^ Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523-530, 1995.9.^ *Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area"(http://docserver.carma.newcastle.edu.au/785/) , Journal of Number Theory 128 (1): 17–48,doi:10.1016/j.jnt.2007.05.005 (http://dx.doi.org/10.1016%2Fj.jnt.2007.05.005) , MR 2382768 (http://www.ams.org/mathscinet-getitem?mr=2382768) , http://docserver.carma.newcastle.edu.au/785/.

10.

External links

Weisstein, Eric W., "Pentagon (http://mathworld.wolfram.com/Pentagon.html) " from MathWorld.Animated demonstration (http://www.mathopenref.com/constinpentagon.html) constructing an inscribedpentagon with compass and straightedge.How to construct a regular pentagon (http://www.opentutorial.com/Construct_a_pentagon) with only acompass and straightedge.How to fold a regular pentagon (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#knot) using only a strip of paperDefinition and properties of the pentagon (http://www.mathopenref.com/pentagon.html) , withinteractive animation

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Renaissance artists' approximate constructions of regular pentagons (http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1056&bodyId=1245)Pentagon. (http://whistleralley.com/polyhedra/pentagon.htm) How to calculate various dimensions ofregular pentagons.

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