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OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, an equilateral triangle is a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
in which all three sides have the same length. In the familiar
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, an equilateral triangle is also equiangular; that is, all three internal
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s are also congruent to each other and are each 60°. It is also a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
, so it is also referred to as a regular triangle.

# Principal properties

Denoting the common length of the sides of the equilateral triangle as $a$, we can determine using the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
that: *The area is $A=\frac a^2$, *The perimeter is $p=3a\,\!$ *The radius of the
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
is $R = \frac$ *The radius of the
inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
is $r=\frac a$ or $r=\frac$ *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is $h=\frac a$ Denoting the radius of the circumscribed circle as ''R'', we can determine using
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
that: *The area of the triangle is $\mathrm=\fracR^2$ Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side: *The area is $A=\frac$ *The height of the center from each side, or apothem, is $\frac$ *The radius of the circle circumscribing the three vertices is $R=\frac$ *The radius of the inscribed circle is $r=\frac$ In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.

# Characterizations

A triangle ''ABC'' that has the sides ''a'', ''b'', ''c'',
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate nam ...
''s'',
area Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...
''T'', exradii ''ra'', ''rb'', ''rc'' (tangent to ''a'', ''b'', ''c'' respectively), and where ''R'' and ''r'' are the radii of the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
and
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
respectively, is equilateral
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.

## Sides

*$a=b=c$ *$\frac+\frac+\frac=\frac$

## Semiperimeter

*$s = 2R + \left\left(3\sqrt - 4\right\right)r$ (Blundon) *$s^2=3r^2+12Rr$ *$s^2=3\sqrtT$ *$s=3\sqrtr$ *$s=\fracR$

## Angles

*$A=B=C=60^\circ$ *$\cos+\cos+\cos=\frac$ *$\sin\sin\sin=\frac$

## Area

*$T=\frac\quad$ ( Weitzenböck) *$T=\frac\left(abc\right)^\frac$

*$R=2r$ (Chapple-Euler) *$9R^2=a^2+b^2+c^2$ *$r=\frac$ *$r_a=r_b=r_c$

## Equal cevians

Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles: *The three altitudes have equal lengths. *The three
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
s have equal lengths. *The three angle bisectors have equal lengths.

## Coincident triangle centers

Every
triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...
of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, incl ...
connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular: *A triangle is equilateral if any two of the circumcenter,
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bise ...
, centroid, or
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ...
coincide. *It is also equilateral if its circumcenter coincides with the
Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concu ...
, or if its incenter coincides with its
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle t ...
.

## Six triangles formed by partitioning by the medians

For any triangle, the three medians partition the triangle into six smaller triangles. *A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius. *A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.

## Points in the plane

*A triangle is equilateral if and only if, for ''every'' point ''P'' in the plane, with distances ''p'', ''q'', and ''r'' to the triangle's sides and distances ''x'', ''y'', and ''z'' to its vertices, $4\left(p^2 + q^2 + r^2\right) \geq x^2 + y^2 + z^2.$

# Notable theorems

Morley's trisector theorem In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theore ...
states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle. A version of the
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
for triangles states that the triangle of greatest
area Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...
among all those with a given
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pra ...
is equilateral. Viviani's theorem states that, for any interior point ''P'' in an equilateral triangle with distances ''d'', ''e'', and ''f'' from the sides and altitude ''h'', $d+e+f = h,$ independent of the location of ''P''. Pompeiu's theorem states that, if ''P'' is an arbitrary point in the plane of an equilateral triangle ''ABC'' but not on its
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, then there exists a triangle with sides of lengths ''PA'', ''PB'', and ''PC''. That is, ''PA'', ''PB'', and ''PC'' satisfy the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
that the sum of any two of them is greater than the third. If ''P'' is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem.

# Other properties

By Euler's inequality, the equilateral triangle has the smallest ratio ''R''/''r'' of the circumradius to the inradius of any triangle: specifically, ''R''/''r'' = 2. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, $\frac$, is larger than that of any non-equilateral triangle. The ratio of the area to the square of the perimeter of an equilateral triangle, $\frac,$ is larger than that for any other triangle.Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas ''A''1 and ''A''2, then $\frac \leq \frac \leq \frac.$ If a triangle is placed in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
with complex vertices ''z''1, ''z''2, and ''z''3, then for either non-real cube root $\omega$ of 1 the triangle is equilateral if and only if $z_1 + \omega z_2 + \omega^2 z_3 = 0.$ Given a point ''P'' in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when ''P'' is the centroid. In no other triangle is there a point for which this ratio is as small as 2. This is the Erdős–Mordell inequality; a stronger variant of it is Barrow's inequality, which replaces the perpendicular distances to the sides with the distances from ''P'' to the points where the angle bisectors of ∠''APB'', ∠''BPC'', and ∠''CPA'' cross the sides (''A'', ''B'', and ''C'' being the vertices). For any point ''P'' in the plane, with distances ''p'', ''q'', and ''t'' from the vertices ''A'', ''B'', and ''C'' respectively, $3 \left(p^4 + q^4 + t^4 + a^4\right) = \left(p^2 + q^2 + t^2 + a^2\right)^2.$ For any point ''P'' in the plane, with distances ''p'', ''q'', and ''t'' from the vertices, $p^2+q^2+t^2 = 3\left(R^2 + L^2\right)$ and where ''R'' is the circumscribed radius and ''L'' is the distance between point ''P'' and the centroid of the equilateral triangle. For any point ''P'' on the inscribed circle of an equilateral triangle, with distances ''p'', ''q'', and ''t'' from the vertices, $4 \left(p^2 + q^2 + t^2\right) = 5a^2$ and $16 \left(p^4 + q^4 + t^4\right) = 11a^4.$ For any point ''P'' on the minor arc BC of the circumcircle, with distances ''p'', ''q'', and ''t'' from A, B, and C respectively, $p = q + t$ and $q^2 + qt + t^2=a^2 ;$ moreover, if point D on side BC divides PA into segments PD and DA with DA having length ''z'' and PD having length ''y'', then $z = \frac,$ which also equals $\tfrac$ if ''t'' ≠ ''q''; and $\frac+\frac=\frac,$ which is the
optic equation In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':Dickson, L. E., ''History of the Theory of Numbers, ...
. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral. An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
of order 3 about its center. Its
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
is the
dihedral group of order 6 In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abel ...
''D''3. Equilateral triangles are the only triangles whose
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse insc ...
is a circle (specifically, it is the incircle). The integer-sided equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees.Conway, J. H., and Guy, R. K., "The only rational triangle", in ''The Book of Numbers'', 1996, Springer-Verlag, pp. 201 and 228–239. The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the
heptagonal triangle A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter an ...
being the only obtuse one).Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", ''
Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
'' 46 (1), January 1973, 7–19.
The equilateral triangle can be inscribed inside any other regular polygon, including itself, with the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
being the only other regular polygon with this property. The equilateral triangle tiles two dimensional space, with six triangles meeting at a vertex. It has a regular dual tessellation, the
hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathematic ...
. 3.122, 3.4.6.4, (3.6)2, 32.4.3.4, and 34.6 are all semi-regular tessellations constructed with 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. In three dimensions, they form faces of regular and uniform
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
. Three of the five
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...
s are composed of equilateral triangles: the
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
,
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...
. In particular, the tetrahedron, which has four equilateral triangles for faces, can be considered the three-dimensional analogue of the triangle. All Platonic solids can inscribe tetrahedra, as well as be inscribed inside tetrahedra. Also in the third dimension, equilateral triangles form uniform antiprisms as well as uniform star antiprisms. For antiprisms, two (non-mirrored) parallel copies of regular polygons are connected by alternating bands of 2''n'' triangles. Specifically for star antiprisms, there are prograde and retrograde (crossed) solutions that join mirrored and non-mirrored parallel star polygons. The equilateral triangle belongs to the infinite family of ''n''- simplexes, with ''n''=2.

# Geometric construction

An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment An alternative method is to 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 centers of the circles and either of the points of intersection. In both methods a by-product is the formation of
vesica piscis The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. In Latin, "vesica piscis" literal ...
. The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of Euclid's ''Elements''.

# Derivation of area formula

The area formula $A = \fraca^2$ in terms of side length ''a'' can be derived directly using the Pythagorean theorem or using trigonometry.

## Using the Pythagorean theorem

The area of a triangle is half of one side ''a'' times the height ''h'' from that side: $A = \frac ah.$ The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base ''a'', and the hypotenuse is the side ''a'' of the equilateral triangle. The height of an equilateral triangle can be found using the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
$\left(\frac\right)^2 + h^2 = a^2$ so that $h = \fraca.$ Substituting ''h'' into the area formula ''ah'' gives the area formula for the equilateral triangle: $A = \fraca^2.$

## Using trigonometry

Using
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, the area of a triangle with any two sides ''a'' and ''b'', and an angle ''C'' between them is $A = \frac ab \sin C.$ Each angle of an equilateral triangle is 60°, so $A = \frac ab \sin 60^\circ.$ The sine of 60° is $\tfrac$. Thus $A = \frac ab \times \frac = \fracab = \fraca^2$ since all sides of an equilateral triangle are equal.

# In culture and society

Equilateral triangles have frequently appeared in man made constructions: *The shape occurs in modern architecture such as the cross-section of the
Gateway Arch The Gateway Arch is a monument in St. Louis, Missouri, United States. Clad in stainless steel and built in the form of a weighted catenary arch, it is the world's tallest arch and Missouri's tallest accessible building. Some sources consider ...
. *Its applications in flags and heraldry includes the flag of Nicaragua and the
flag of the Philippines The national flag of the Philippines ( tgl, Pambansang watawat ng Pilipinas; ilo, Nailian a bandera ti Filipinas; ceb, Nasudnong bandila ng Pilipinas; es, Bandera Nacional de Filipinas) is a horizontal bicolor flag with equal bands of ro ...
. *It is a shape of a variety of road signs, including the
yield sign In road transport, a yield or give way sign indicates that merging drivers must prepare to stop if necessary to let a driver on another approach proceed. A driver who stops or slows down to let another vehicle through has yielded the right of ...
.