# Dictionary Definition

triangular adj

1 having three angles; forming or shaped like a
triangle; "a triangular figure"; "a triangular pyrimid has a
triangle for a base"

2 involving three parties or elements; "the
triangular mother-father-child relationship"; "a trilateral
agreement"; "a tripartite treaty"; "a tripartite division"; "a
three-way playoff" [syn: trilateral, tripartite, three-party,
three-way]

3 having three sides; "a trilateral figure" [syn:
trilateral, three-sided]

# User Contributed Dictionary

## English

### Adjective

- Of, or pertaining to, triangles.
- Shaped like a triangle.
- having a triangle as a base; as, a triangular prism, a triangular pyramid.
- having three elements or parties; trilateral, tripartite.

#### Related terms

#### Translations

- Danish: trekantet
- German: dreieckig
- Latin: triangulus
- Swedish: triangulärt

# Extensive Definition

A triangle is one of the basic shapes of geometry: a polygon with three corners or
vertices and
three sides or edges which are line
segments. A triangle with vertices A, B, and C is denoted
trianglenotation ABC.

In Euclidean
geometry any three non-collinear points determine a
unique triangle and a unique plane
(i.e. two-dimensional Cartesian
space).

## Types of triangles

Triangles can be classified according to the
relative lengths of their sides:

- In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon.
- In an isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two). An isosceles triangle also has two equal angles: the angles opposite the two equal sides.
- In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.

EquilateralIsoscelesScalene

Triangles can also be classified according to
their internal angles, described below using degrees of
arc:

- A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
- An oblique triangle has no internal angle equal to 90°.
- An obtuse triangle is an oblique triangle with one internal angle larger than 90° (an obtuse angle).
- An acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.

RightObtuseAcute

\underbrace_

Oblique

## Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.The angles of a triangle add up to 180 degrees.
An exterior
angle of a triangle (an angle that is adjacent and
supplementary to an internal angle) is always equal to the two
angles of a triangle that it is not adjacent/supplementary to. Like
all convex polygons, the
exterior angles of a triangle add up to 360 degrees.

The sum of the lengths of any two sides of a
triangle always exceeds the length of the third side. That is the
triangle
inequality. (In the special case of equality, two of the angles
have collapsed to size zero, and the triangle has degenerated to a
line segment.)

Two triangles are said to be similar
if and only if the angles of one are equal to the corresponding
angles of the other. In this case, the lengths of their
corresponding sides are
proportional. This occurs for example when two triangles share
an angle and the sides opposite to that angle are parallel.

A few basic postulates and theorems about similar
triangles:

- Two triangles are similar if at least two corresponding angles are equal.
- If two corresponding sides of two triangles are in proportion, and their included angles are equal, the triangles are similar.
- If three sides of two triangles are in proportion, the triangles are similar.

For two triangles to be congruent, each of their
corresponding angles and sides must be equal (6 total). A few basic
postulates and theorems about congruent triangles:

- SAS Postulate: If two sides and the included angles of two triangles are correspondingly equal, the two triangles are congruent.
- SSS Postulate: If every side of two triangles are correspondingly equal, the triangles are congruent.
- ASA Postulate: If two angles and the included sides of two triangles are correspondingly equal, the two triangles are congruent.
- AAS Theorem: If two angles and any side of two triangles are correspondingly equal, the two triangles are congruent.
- Hypotenuse-Leg Theorem: If the hypotenuses and one leg of two right triangles are correspondingly equal, the triangles are congruent.

Using right triangles and the concept of
similarity, the trigonometric
functions sine and cosine can be defined. These are functions
of an angle which are
investigated in trigonometry.

In Euclidean geometry, the sum of the internal
angles of a triangle is equal to 180°. This allows determination of
the third angle of any triangle as soon as two angles are
known.

A central theorem is the Pythagorean
theorem, which states in any right triangle, the square of the
length of the hypotenuse equals the sum of
the squares of the lengths of the two other sides. If the
hypotenuse has length c, and the legs have lengths a and b, then
the theorem states that

- a^2 + b^2=c^2. \,

The converse is true: if the lengths of the sides
of a triangle satisfy the above equation, then the triangle is a
right triangle.

Some other facts about right triangles:

- The acute angles of a right triangle are complementary.
- If the legs of a right triangle are equal, then the angles opposite the legs are equal, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times the square root of two.
- In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side.
- In all right triangles, the median on the hypotenuse is the half of the hypotenuse.

## Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.A perpendicular bisector of a
triangle is a straight line passing through the midpoint of a side
and being perpendicular to it, i.e. forming a right angle with it.
The three perpendicular bisectors meet in a single point, the
triangle's circumcenter; this point is
the center of the circumcircle, the circle passing through all three
vertices. The diameter of this circle can be found from the law of
sines stated above.

Thales'
theorem implies that if the circumcenter is located on one side
of the triangle, then the opposite angle is a right one. More is
true: if the circumcenter is located inside the triangle, then the
triangle is acute; if the circumcenter is located outside the
triangle, then the triangle is obtuse.

An altitude
of a triangle is a straight line through a vertex and perpendicular
to (i.e. forming a right angle with) the opposite side. This
opposite side is called the base of the altitude, and the point
where the altitude intersects the base (or its extension) is called
the foot of the altitude. The length of the altitude is the
distance between the base and the vertex. The three altitudes
intersect in a single point, called the orthocenter of the triangle.
The orthocenter lies inside the triangle if and only if the
triangle is acute. The three vertices together with the orthocenter
are said to form an orthocentric
system.

An angle
bisector of a triangle is a straight line through a vertex
which cuts the corresponding angle in half. The three angle
bisectors intersect in a single point, the incenter, the center of the
triangle's incircle.
The incircle is the circle which lies inside the triangle and
touches all three sides. There are three other important circles,
the excircles; they lie
outside the triangle and touch one side as well as the extensions
of the other two. The centers of the in- and excircles form an
orthocentric
system.

A median
of a triangle is a straight line through a vertex and the midpoint
of the opposite side, and divides the triangle into two equal
areas. The three medians intersect in a single point, the
triangle's centroid.
This is also the triangle's center of
gravity: if the triangle were made out of wood, say, you could
balance it on its centroid, or on any line through the centroid.
The centroid cuts every median in the ratio 2:1, i.e. the distance
between a vertex and the centroid is twice as large as the distance
between the centroid and the midpoint of the opposite side.

The midpoints of the three sides and the feet of
the three altitudes all lie on a single circle, the triangle's
nine-point
circle. The remaining three points for which it is named are
the midpoints of the portion of altitude between the vertices and
the orthocenter. The
radius of the nine-point circle is half that of the circumcircle.
It touches the incircle (at the Feuerbach
point) and the three excircles.

The centroid (yellow), orthocenter (blue),
circumcenter (green) and barycenter of the nine-point circle (red
point) all lie on a single line, known as Euler's line
(red line). The center of the nine-point circle lies at the
midpoint between the orthocenter and the circumcenter, and the
distance between the centroid and the circumcenter is half that
between the centroid and the orthocenter.

The center of the incircle is not in general
located on Euler's line.

If one reflects a median at the angle bisector
that passes through the same vertex, one obtains a symmedian. The three
symmedians intersect in a single point, the symmedian
point of the triangle.

## Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known, and simplest formula is- S=\fracbh

Although simple, this formula is only useful if
the height can be readily found. For example, the surveyor of a
triangular field measures the length of each side, and can find the
area from his results without having to construct a 'height'.
Various methods may be used in practice, depending on what is known
about the triangle. The following is a selection of frequently used
formulae for the area of a triangle.

### Using vectors

The area of a parallelogram can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then |AB × AC|, which is the magnitude of the cross product of vectors AB and AC. |AB × AC| is equal to |h × AC|, where h represents the altitude h as a vector.The area of triangle ABC is half of this, or
S = ½|AB × AC|.

The area of triangle ABC can also be expressed in
term of dot products
as follows:

\frac \sqrt =\frac \sqrt \, .

### Using trigonometry

The height of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as:- S = \fracab\sin \gamma = \fracbc\sin \alpha = \fracca\sin \beta.

Furthermore, since sin α = sin (π - α) = sin (β +
γ), and similarly for the other two angles:

- S = \fracab\sin (\alpha+\beta) = \fracbc\sin (\beta+\gamma) = \fracca\sin (\gamma+\alpha).

### Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area S can be computed as ½ times the absolute value of the determinant- S=\frac\left|\det\beginx_B & x_C \\ y_B & y_C \end\right| = \frac|x_B y_C - x_C y_B|.

For three general vertices, the equation
is:

- S=\frac \left| \det\beginx_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end \right| = \frac \big| x_A y_C - x_A y_B + x_B y_A - x_B y_C + x_C y_B - x_C y_A \big|.

In three dimensions, the area of a general
triangle is the Pythagorean
sum of the areas of the respective projections on the three
principal planes (i.e. x = 0, y = 0 and z = 0):

- S=\frac \sqrt.

### Using Heron's formula

The shape of the triangle is determined by the lengths of the sides alone. Therefore the area S also can be derived from the lengths of the sides. By Heron's formula:- S = \sqrt

where
s = ½ (a + b + c)
is the semiperimeter, or half of the triangle's perimeter.

An equivalent way of writing Heron's formula
is

- S = \frac \sqrt.

## Non-planar triangles

A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.While all regular, planar (two dimensional)
triangles contain angles that add up to 180°, there are cases in
which the angles of a triangle can be greater than or less than
180°. In curved figures, a triangle on a negatively curved figure
("saddle") will have its angles add up to less than 180° while a
triangle on a positively curved figure ("sphere") will have its
angles add up to more than 180°. Thus, if one were to draw a giant
triangle on the surface of the Earth, one would find that the sum
of its angles were greater than 180°.

## See also

- List of triangle topics
- Triangular number
- Special right triangles
- Fermat point
- Hadwiger-Finsler inequality
- Pedoe's inequality
- Ono's inequality
- Lester's theorem
- Congruence (geometry)
- Pythagorean theorem
- Law of sines
- Law of cosines
- Law of tangents
- Triangulation (topology) of a manifold
- Triangulated category
- The inertia tensor of a triangle in three-dimensional space

## References

## External links

- Triangle Formulas at Geometry Atlas.
- Triangle Calculator - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.
- Clark Kimberling: Encyclopedia of triangle centers. Lists some 3200 interesting points associated with any triangle.
- Christian Obrecht: Eukleides. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.
- Proof that the sum of the angles in a triangle is 180 degrees
- Area of a triangle - 7 different ways
- Triangle definition pages with interactive applets that are also useful in a classroom setting.
- Animated demonstrations of triangle constructions using compass and straightedge.
- Triangles: Theorems and Problems. Interactive illustrations at Geometry from the Land of the Incas.
- Many things about triangles
- Triangle in light aircraft
- All Math Words Encyclopedia for grades 7-10.

triangular in Contenese: 三角形

triangular in Arabic: مثلث

triangular in Aragonese: Trianglo

triangular in Asturian: Triángulu

triangular in Aymara: Mujina

triangular in Azerbaijani: Üçbucaqlar

triangular in Bengali: ত্রিভুজ

triangular in Min Nan: Saⁿ-kak-hêng

triangular in Bulgarian: Триъгълник

triangular in Catalan: Triangle

triangular in Chuvash: Виç кĕтеслĕх

triangular in Czech: Trojúhelník

triangular in Corsican: Triangulu

triangular in Welsh: Triongl

triangular in Danish: Trekant

triangular in German: Dreieck

triangular in Estonian: Kolmnurk

triangular in Modern Greek (1453-):
Τρίγωνο

triangular in Spanish: Triángulo

triangular in Esperanto: Triangulo

triangular in Basque: Hiruki

triangular in Persian: مثلث

triangular in French: Triangle

triangular in Galician: Triángulo

triangular in Classical Chinese: 三角形

triangular in Korean: 삼각형

triangular in Croatian: Trokut

triangular in Ido: Triangulo

triangular in Indonesian: Segitiga

triangular in Icelandic: Þríhyrningur

triangular in Italian: Triangolo

triangular in Hebrew: משולש

triangular in Georgian: სამკუთხედი

triangular in Central Khmer: ត្រីកោណ

triangular in Swahili (macrolanguage):
Pembetatu

triangular in Haitian: Triyang

triangular in Kurdish: Sêgoşe

triangular in Latin: Triangulum

triangular in Latvian: Trīsstūris

triangular in Lithuanian: Trikampis

triangular in Limburgan: Driehook

triangular in Hungarian: Háromszög

triangular in Macedonian: Триаголник

triangular in Malayalam: ത്രികോണം

triangular in Marathi: त्रिकोण

triangular in Malay (macrolanguage): Segi
tiga

triangular in Mongolian: Гурвалжин

triangular in Dutch: Driehoek (meetkunde)

triangular in Newari: त्रिकोण

triangular in Japanese: 三角形

triangular in Norwegian: Trekant

triangular in Norwegian Nynorsk: Trekant

triangular in Narom: Trian

triangular in Polish: Trójkąt

triangular in Portuguese: Triângulo

triangular in Romanian: Triunghi

triangular in Quechua: Kimsak'uchu

triangular in Russian: Треугольник

triangular in Scots: Triangle

triangular in Simple English: Triangle

triangular in Slovak: Trojuholník

triangular in Slovenian: Trikotnik

triangular in Serbian: Троугао

triangular in Serbo-Croatian: Trokut

triangular in Finnish: Kolmio

triangular in Swedish: Triangel

triangular in Tamil: முக்கோணம்

triangular in Telugu: త్రిభుజం

triangular in Thai: รูปสามเหลี่ยม

triangular in Vietnamese: Tam giác

triangular in Tajik: Секунҷа

triangular in Turkish: Üçgen

triangular in Ukrainian: Трикутник

triangular in Urdu: مثلث

triangular in Vlaams: Drieoek

triangular in Yiddish: דרייעק

triangular in Yoruba: Anígunmẹ́ta

triangular in Chinese: 三角形

# Synonyms, Antonyms and Related Words

cuneate, cuneated, cuneiform, deltoid, fan-shaped, oxygonal, three, three-cornered,
three-dimensional, three-footed, three-in-one, three-pronged,
three-sided, triadic,
trial, triangulate, triarch, trichotomous, tricorn, tricornered, tricuspid, trident, tridental, trifid, triflorate, triflorous, trifurcate, trigonal, trigonoid, trigrammatic, trihedral, trilateral, trimerous, trinal, trine, triparted, tripartite, triple, triplex, tripodic, triquetral, triquetrous, trisected, triune