# Equilateral triangle

Last updated
Equilateral triangle
Type Regular polygon
Edges and vertices 3
Schläfli symbol {3}
Coxeter–Dynkin diagrams
Symmetry group D3
Area ${\displaystyle {\tfrac {\sqrt {3}}{4}}a^{2}}$
Internal angle (degrees)60°

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

## Principal properties

Denoting the common length of the sides of the equilateral triangle as ${\displaystyle a}$, we can determine using the Pythagorean theorem that:

• The area is ${\displaystyle A={\frac {\sqrt {3}}{4}}a^{2},}$
• The perimeter is ${\displaystyle p=3a\,\!}$
• The radius of the circumscribed circle is ${\displaystyle R={\frac {a}{\sqrt {3}}}}$
• The radius of the inscribed circle is ${\displaystyle r={\frac {\sqrt {3}}{6}}a}$ or ${\displaystyle r={\frac {R}{2}}}$
• The geometric center of the triangle is the center of the circumscribed and inscribed circles
• The altitude (height) from any side is ${\displaystyle h={\frac {\sqrt {3}}{2}}a}$

Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:

• The area of the triangle is ${\displaystyle A={\frac {3{\sqrt {3}}}{4}}R^{2}}$

Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:

• The area is ${\displaystyle A={\frac {h^{2}}{\sqrt {3}}}}$
• The height of the center from each side, or apothem, is ${\displaystyle {\frac {h}{3}}}$
• The radius of the circle circumscribing the three vertices is ${\displaystyle R={\frac {2h}{3}}}$
• The radius of the inscribed circle is ${\displaystyle r={\frac {h}{3}}}$

In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.

## Characterizations

A triangle ${\displaystyle ABC}$ that has the sides ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, semiperimeter ${\displaystyle s}$, area ${\displaystyle T}$, exradii ${\displaystyle r_{a}}$, ${\displaystyle r_{b}}$, ${\displaystyle r_{c}}$ (tangent to ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ respectively), and where ${\displaystyle R}$ and ${\displaystyle r}$ are the radii of the circumcircle and incircle respectively, is equilateral if and only if 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

• ${\displaystyle a=b=c}$
• ${\displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}}$ [1]

### Semiperimeter

• ${\displaystyle s=2R+\left(3{\sqrt {3}}-4\right)r}$ [2] (Blundon)
• ${\displaystyle s^{2}=3r^{2}+12Rr}$ [3]
• ${\displaystyle s^{2}=3{\sqrt {3}}T}$ [4]
• ${\displaystyle s=3{\sqrt {3}}r}$
• ${\displaystyle s={\frac {3{\sqrt {3}}}{2}}R}$

### Angles

• ${\displaystyle A=B=C=60^{\circ }}$
• ${\displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}}$
• ${\displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}}$ [5]

### Area

• ${\displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad }$ (Weitzenböck)
• ${\displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}}$ [4]

• ${\displaystyle R=2r}$ [6] (Chapple-Euler)
• ${\displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}}$ [6]
• ${\displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}}$ [5]
• ${\displaystyle r_{a}=r_{b}=r_{c}}$

### Equal cevians

Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles: [7]

### Coincident triangle centers

Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line 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:

### 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. [9] :Theorem 1
• A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid. [9] :Corollary 7

### Points in the plane

• A triangle is equilateral if and only if, for every point ${\displaystyle P}$ in the plane, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle r}$ to the triangle's sides and distances ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ to its vertices, [10] :p.178,#235.4
${\displaystyle 4\left(p^{2}+q^{2}+r^{2}\right)\geq x^{2}+y^{2}+z^{2}.}$

## Notable theorems

Morley's trisector theorem 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 for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. [11]

Viviani's theorem states that, for any interior point ${\displaystyle P}$ in an equilateral triangle with distances ${\displaystyle d}$, ${\displaystyle e}$, and ${\displaystyle f}$ from the sides and altitude ${\displaystyle h}$,

${\displaystyle d+e+f=h,}$

independent of the location of ${\displaystyle P}$. [12]

Pompeiu's theorem states that, if ${\displaystyle P}$ is an arbitrary point in the plane of an equilateral triangle ${\displaystyle ABC}$ but not on its circumcircle, then there exists a triangle with sides of lengths ${\displaystyle PA}$, ${\displaystyle PB}$, and ${\displaystyle PC}$. That is, ${\displaystyle PA}$, ${\displaystyle PB}$, and ${\displaystyle PC}$ satisfy the triangle inequality that the sum of any two of them is greater than the third. If ${\displaystyle P}$ 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.

## 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 ${\displaystyle r}$, 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 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 ${\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}}$ in terms of side length ${\displaystyle a}$ 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 ${\displaystyle a}$ times the height ${\displaystyle h}$ from that side:

${\displaystyle A={\frac {1}{2}}ah.}$

The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base ${\displaystyle a}$, and the hypotenuse is the side ${\displaystyle a}$ of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem

${\displaystyle \left({\frac {a}{2}}\right)^{2}+h^{2}=a^{2}}$

so that

${\displaystyle h={\frac {\sqrt {3}}{2}}a.}$

Substituting ${\displaystyle h}$ into the area formula ${\displaystyle {\frac {1}{2}}ah}$ gives the area formula for the equilateral triangle:

${\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}.}$

### Using trigonometry

Using trigonometry, the area of a triangle with any two sides ${\displaystyle a}$ and ${\displaystyle b}$, and an angle ${\displaystyle C}$ between them is

${\displaystyle A={\frac {1}{2}}ab\sin C.}$

Each angle of an equilateral triangle is 60°, so

${\displaystyle A={\frac {1}{2}}ab\sin 60^{\circ }.}$

The sine of 60° is ${\displaystyle {\tfrac {\sqrt {3}}{2}}}$. Thus

${\displaystyle A={\frac {1}{2}}ab\times {\frac {\sqrt {3}}{2}}={\frac {\sqrt {3}}{4}}ab={\frac {\sqrt {3}}{4}}a^{2}}$

since all sides of an equilateral triangle are equal.

## Other properties

An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center, whose symmetry group is the dihedral group of order 6, ${\displaystyle \mathrm {D} _{3}}$. The integer-sided equilateral triangle is the only triangle with integer sides, and three rational angles as measured in degrees. [13] It is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes), [14] :p. 19 and the only triangle whose Steiner inellipse is a circle (specifically, the incircle). 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 also equilateral. [15] It is the only regular polygon aside from the square that can be inscribed inside any other regular polygon.

By Euler's inequality, the equilateral triangle has the smallest ratio of the circumradius ${\displaystyle R}$ to the inradius ${\displaystyle r}$ of any triangle, with [16] :p.198

${\displaystyle {\frac {R}{r}}=2.}$

Given a point ${\displaystyle P}$ 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 ${\displaystyle P}$ is the centroid. In no other triangle is there a point for which this ratio is as small as 2. [17] 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 ${\displaystyle P}$ to the points where the angle bisectors of ${\displaystyle \angle APB}$, ${\displaystyle \angle BPC}$, and ${\displaystyle \angle CPA}$ cross the sides (${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ being the vertices). There are numerous other triangle inequalities that hold with equality if and only if the triangle is equilateral.

For any point ${\displaystyle P}$ in the plane, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from the vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ respectively, [18]

${\displaystyle 3\left(p^{4}+q^{4}+t^{4}+a^{4}\right)=\left(p^{2}+q^{2}+t^{2}+a^{2}\right)^{2}.}$

For any point ${\displaystyle P}$ in the plane, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from the vertices, [19]

${\displaystyle p^{2}+q^{2}+t^{2}=3\left(R^{2}+L^{2}\right),}$
${\displaystyle p^{4}+q^{4}+t^{4}=3\left[\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right],}$

where ${\displaystyle R}$ is the circumscribed radius and ${\displaystyle L}$ is the distance between point ${\displaystyle P}$ and the centroid of the equilateral triangle.

For any point ${\displaystyle P}$ on the inscribed circle of an equilateral triangle, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from the vertices, [20]

${\displaystyle 4\left(p^{2}+q^{2}+t^{2}\right)=5a^{2},}$
${\displaystyle 16\left(p^{4}+q^{4}+t^{4}\right)=11a^{4}.}$

For any point ${\displaystyle P}$ on the minor arc ${\displaystyle BC}$ of the circumcircle, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, respectively [12]

${\displaystyle p=q+t,}$
${\displaystyle q^{2}+qt+t^{2}=a^{2}.}$

Moreover, if point ${\displaystyle D}$ on side ${\displaystyle BC}$ divides ${\displaystyle PA}$ into segments ${\displaystyle PD}$ and ${\displaystyle DA}$ with ${\displaystyle DA}$ having length ${\displaystyle z}$ and ${\displaystyle PD}$ having length ${\displaystyle y}$, then [12] :172

${\displaystyle z={\frac {t^{2}+tq+q^{2}}{t+q}},}$

which also equals ${\textstyle {\tfrac {t^{3}-q^{3}}{t^{2}-q^{2}}}}$ if ${\displaystyle t\neq q}$ and

${\displaystyle {\frac {1}{q}}+{\frac {1}{t}}={\frac {1}{y}},}$

which is the optic equation.

For an equilateral triangle:

• The ratio of its area to the area of the incircle, ${\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}$, is the largest of any triangle. [21] :Theorem 4.1
• The ratio of its area to the square of its perimeter, ${\displaystyle {\frac {1}{12{\sqrt {3}}}},}$ is larger than that of any non-equilateral triangle. [11]
• If a segment splits an equilateral triangle into two regions with equal perimeters and with areas ${\displaystyle A_{1}}$ and ${\displaystyle A_{2}}$, then [10] :p.151,#J26
${\displaystyle {\frac {7}{9}}\leq {\frac {A_{1}}{A_{2}}}\leq {\frac {9}{7}}.}$

If a triangle is placed in the complex plane with complex vertices ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, and ${\displaystyle z_{3}}$, then for either non-real cube root ${\displaystyle \omega }$ of 1 the triangle is equilateral if and only if [22] :Lemma 2

${\displaystyle z_{1}+\omega z_{2}+\omega ^{2}z_{3}=0.}$

Notably, the equilateral triangle tiles two dimensional space with six triangles meeting at a vertex, whose dual tessellation is the hexagonal tiling. 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. [23]

In three dimensions, equilateral triangles form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles: the tetrahedron, octahedron and icosahedron. [24] :p.238 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. Equilateral triangles also form uniform antiprisms as well as uniform star antiprisms in three-dimensional space. For antiprisms, two (non-mirrored) parallel copies of regular polygons are connected by alternating bands of ${\displaystyle 2n}$ equilateral triangles. [25] Specifically for star antiprisms, there are prograde and retrograde (crossed) solutions that join mirrored and non-mirrored parallel star polygons. [26] [27] The Platonic octahedron is also a triangular antiprism, which is the first true member of the infinite family of antiprisms (the tetrahedron, as a digonal antiprism, is sometimes considered the first). [24] :p.240

As a generalization, the equilateral triangle belongs to the infinite family of ${\displaystyle n}$-simplexes, with ${\displaystyle n=2}$. [28]

## In culture and society

Equilateral triangles have frequently appeared in man made constructions:

## Related Research Articles

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid . Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".

In geometry, an octahedron 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 each vertex.

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.

In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the side opposite the vertex. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape also has alternative names called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron; these names mean the 12-sided polyhedron.

In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center.

In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Algorithms also exist to solve the smallest-circle problem explicitly.

In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. It is named after Paul Erdős and Louis Mordell. Erdős (1935) posed the problem of proving the inequality; a proof was provided two years later by Mordell and D. F. Barrow. This solution was however not very elementary. Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007).

In geometry, the Steiner inellipse, midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inellipse. By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral. The Steiner inellipse is attributed by Dörrie to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

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Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds