Decagon
Regular decagon | |
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Dual polygon | Self |
In geometry, a decagon (from the Greek δέκα déka and γωνία gonía, "ten angles") is a ten-sided polygon or 10-gon.[1] The total sum of the interior angles of a simple decagon is 1440°.
Regular decagon
A regular decagon has all sides of equal length and each internal angle will always be equal to 144°.[1] Its Schläfli symbol is {10} [2] and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.
Side length
The picture shows a regular decagon with side length and radius of the circumscribed circle.
- The triangle has two equally long legs with length and a base with length
- The circle around with radius intersects in a point (not designated in the picture).
- Now the triangle is an isosceles triangle with vertex and with base angles .
- Therefore . So and hence is also an isosceles triangle with vertex . The length of its legs is , so the length of is .
- The isosceles triangles and have equal angles of 36° at the vertex, and so they are similar, hence:
- Multiplication with the denominators leads to the quadratic equation:
- This equation for the side length has one positive solution:
So the regular decagon can be constructed with ruler and compass.
- Further conclusions
and the base height of (i.e. the length of ) is and the triangle has the area: .
Area
The area of a regular decagon of side length a is given by:[3]
In terms of the apothem r (see also inscribed figure), the area is:
In terms of the
An alternative formula is where d is the distance between parallel sides, or the height when the decagon stands on one side as base, or the diameter of the decagon's inscribed circle. By simple trigonometry,
and it can be written algebraically as
Sides
A regular decagon has 10 sides and is
Construction
As 10 = 2 × 5, a
An alternative (but similar) method is as follows:
- Construct a pentagon in a circle by one of the methods shown in constructing a pentagon.
- Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon. In other words, the image of a regular pentagon under a point reflection with respect of its center is a concentric congruent pentagon, and the two pentagons have in total the vertices of a concentric regular decagon.
- The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.
The golden ratio in decagon
Both in the construction with given circumcircle[5] as well as with given side length is the golden ratio dividing a line segment by exterior division the determining construction element.
- In the construction with given circumcircle the circular arc around G with radius GE3 produces the segment AH, whose division corresponds to the golden ratio.
- In the construction with given side length[6] the circular arc around D with radius DA produces the segment E10F, whose division corresponds to the golden ratio.
Symmetry
The regular decagon has
These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[7] Full symmetry of the regular form is r20 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 g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g10 subgroup has no degrees of freedom but can be seen as
The highest symmetry irregular decagons are d10, an isogonal decagon constructed by five mirrors which can alternate long and short edges, and p10, an isotoxal decagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular decagon.
Dissection
10-cube projection | 40 rhomb dissection | |||
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5-cube |
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Skew decagon
{5}#{ } | {5/2}#{ } | {5/3}#{ } |
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A regular skew decagon is seen as zig-zagging edges of a pentagonal antiprism, a pentagrammic antiprism, and a pentagrammic crossed-antiprism. |
A skew decagon is a skew polygon with 10 vertices and edges but not existing on the same plane. The interior of such a decagon is not generally defined. A skew zig-zag decagon has vertices alternating between two parallel planes.
A
These can also be seen in these four convex polyhedra with icosahedral symmetry. The polygons on the perimeter of these projections are regular skew decagons.
Dodecahedron |
Icosahedron |
Icosidodecahedron |
Rhombic triacontahedron |
Petrie polygons
The regular skew decagon is the
A9 | D6 | B5 | ||
---|---|---|---|---|
9-simplex |
411 |
131 |
5-orthoplex |
5-cube |
See also
- Decagonal number and centered decagonal number, figurate numbers modeled on the decagon
- Decagram, a star polygon with the same vertex positions as the regular decagon
References
- ^ ISBN 9780471461630.
- ISBN 9780521098595.
- ^ The elements of plane and spherical trigonometry, Society for Promoting Christian Knowledge, 1850, p. 59. Note that this source uses a as the edge length and gives the argument of the cotangent as an angle in degrees rather than in radians.
- ^ Ludlow, Henry H. (1904), Geometric Construction of the Regular Decagon and Pentagon Inscribed in a Circle, The Open Court Publishing Co..
- ^ a b Green, Henry (1861), Euclid's Plane Geometry, Books III–VI, Practically Applied, or Gradations in Euclid, Part II, London: Simpkin, Marshall,& CO., p. 116. Retrieved 10 February 2016.
- ^ a b Köller, Jürgen (2005), Regelmäßiges Zehneck, → 3. Section "Formeln, Ist die Seite a gegeben ..." (in German). Retrieved 10 February 2016.
- ISBN 978-1-56881-220-5(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
- Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
- ^ Coxeter, Regular polytopes, 12.4 Petrie polygon, pp. 223-226.
External links
- Weisstein, Eric W. "Decagon". MathWorld.
- Definition and properties of a decagon With interactive animation