Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have equal lengths. In traditional or Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also equal to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.

Properties

* The area of an equilateral triangle with sides of length a,! is a^2frac{sqrt{3{4}
*Perimeter is P=3a,!
*Circumscribed circle radius r=afrac{sqrt{3{3}
*Inscribed circle radius r=afrac{sqrt{3{6}
*Altitude is afrac{sqrt{3{2}.These formulas can be derived using the Pythagorean theorem.

An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center.Its symmetry group is the dihedral group of order 6 "D"3.

Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling.

A result finding an equilateral triangle associated to any triangle is Morley's trisector theorem.

Geometric construction

An equilateral triangle is easily constructed using a compass.Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point past halfway of the line segment. Repeat with the other side of the line.Finally, connect the point where the two arcs intesect with each end of the line segment

Alternate method: Draw a circle with radius "r", place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centres of the circles and either of the points of intersection.

Almost-equilateral Heronian triangles

A Heronian triangle is a triangle with rational sides and rational area. Since the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides, expressed as integers, are of the form "n" − 1, "n", "n" + 1. The first few examples of these almost-equilateral triangles are set forth in the following table.

Subsequent values of "n" can be found by multiplying the last known value by 4, then subtracting the next to the last one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc), as expressed in:q_n = 4q_{n-1} - q_{n-2}.,!This sequence can also be generated from the solutions to the Pell equation "x"² − 3"y"² = 1, which can in turn be derived from the regular continued fraction expansion for √3. [Takeaki Murasaki (2004), [http://zmath.impa.br/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&bi_op=contains&type=pdf&an=02147316&format=complete "On the Heronian Triple (n+1, n, n−1)"] , Sci. Rep. Fac. Educ., Gunma Univ. 52, 9-15.]

In culture and society

Equilateral triangles have frequently appeared in man made constructions:
*Some archaeological sites have equilateral triangles as part of their construction, for example Lepenski Vir in Serbia.
*The shape also occurs in modern architecture such as Randhurst Mall and the Jefferson National Expansion Memorial.
*The Seal of the President of the Philippines and Flag of Junqueirópolis contain equilateral triangles.
*The shape has been given mystical significance, as a representation of the trinity in The Two Babylons and forming part of the tetractys figure used by the Pythagoreans.

ee also

* Trigonometry
* Viviani's theorem

References

External links

* [http://mathworld.wolfram.com/GeometricConstruction.html MathWorld] - an overview of the Euclidean construction of an equilateral triangle


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  • equilateral triangle — UK [ˌiːkwɪlæt(ə)rəl ˈtraɪæŋɡ(ə)l] / US [ˌɪkwɪlætərəl ˈtraɪæŋɡ(ə)l] / US [ˌekwɪlætərəl ˈtraɪæŋɡ(ə)l] noun [countable] Word forms equilateral triangle : singular equilateral triangle plural equilateral triangles maths a triangle whose sides are all …   English dictionary

  • equilateral triangle — triangle having all sides equal …   English contemporary dictionary

  • equilateral triangle — noun a three sided regular polygon • Syn: ↑equiangular triangle • Hypernyms: ↑triangle, ↑trigon, ↑trilateral, ↑regular polygon • Hyponyms: ↑delta …   Useful english dictionary

  • equilateral triangle — noun A triangle having all three sides equal …   Wiktionary

  • equilateral triangle — e|qui|lat|e|ral tri|an|gle [ˌi:kwılætərəl ˈtraıæŋgəl] n technical a ↑triangle whose three sides are all the same length …   Dictionary of contemporary English

  • equilateral triangle — noun (C) technical a triangle (1) whose three sides are all the same length …   Longman dictionary of contemporary English

  • Circle packing in an equilateral triangle — is a packing problem in discrete mathematics where the objective is to pack n unit circles into the smallest possible equilateral triangle. Optimal solutions are known for n < 13 and for any triangular number of circles, and… …   Wikipedia

  • Triangle isocèle — Triangle Pour les articles homonymes, voir Triangle (homonymie) …   Wikipédia en Français

  • Triangle scalène — Triangle Pour les articles homonymes, voir Triangle (homonymie) …   Wikipédia en Français

  • triangle — [ trijɑ̃gl ] n. m. • v. 1270; lat. triangulum 1 ♦ Figure géométrique, polygone plan à trois côtés. Les trois côtés, les trois sommets, les trois angles d un triangle. Triangle quelconque, scalène, isocèle, équilatéral. Triangle rectangle, qui a… …   Encyclopédie Universelle

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