- Equilateral triangle
In

geometry , an**equilateral triangle**is a triangle in which all three sides have equal lengths. In traditional orEuclidean geometry , equilateral triangles are also equiangular; that is, all three internal angles are also equal to each other and are each 60°. They areregular polygon s, 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 thePythagorean theorem .An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and

rotational symmetry of order 3 about its center.Itssymmetry group is thedihedral 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 solid s 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 thetriangular 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 anirrational 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:

*Somearchaeological site s have equilateral triangles as part of their construction, for exampleLepenski Vir in Serbia.

*The shape also occurs in modern architecture such asRandhurst Mall and theJefferson National Expansion Memorial .

*TheSeal of the President of the Philippines andFlag of Junqueirópolis contain equilateral triangles.

*The shape has been given mystical significance, as a representation of thetrinity inThe Two Babylons and forming part of thetetractys 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

*Wikimedia Foundation.
2010.*