Equilateral triangle

Equilateral triangle

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 <math>a\,\!</math> is <math>\frac{a^2\sqrt{3}}{4}</math> and its altitude is <math>\frac{a\sqrt{3}}{2}</math>. 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 D₃.

File:Viervlak-frame.jpg

Regular tetrahedron is made of four equilateral triangles.

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

File:Equilateral triangle construction.svg

Construction of equilateral triangle with compass and straightedge

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.

Side length			Area
n − 1	n	n + 1
3	4	5	6
13	14	15	84
51	52	53	1170
193	194	195	16296

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

<math>q_n = 4q_{n-1} - q_{n-2}.\,\!</math>

This sequence can also be generated from the solutions to the Pell equation x² − 3y² = 1, which can in turn be derived from the regular continued fraction expansion for √3.

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.

See also

• Trigonometry
• Viviani's theorem

External links

• MathWorld - an overview of the Euclidean construction of an equilateral triangle