A triangle is a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

with three corners and three sides, one of the basic

shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...

s in

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

. The corners, also called ''vertices'', are zero-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

while the sides connecting them, also called ''edges'', are one-dimensional

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

s. A triangle has three

internal angle In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...

s, each one bounded by a pair of adjacent edges; the

sum of angles of a triangle In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, radians, two right angles, or a half- turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. The sum can b ...

always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a

planar region In Euclidean geometry, a plane is a Flat space, flat two-dimensional surface (geometry), surface that extends indefinitely. Euclidean planes often arise as Euclidean subspace, subspaces of three-dimensional space \mathbb^3. A prototypical examp ...

. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The

area of a triangle In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is T=bh/2, where ''b'' is the length of the ''base'' of the triangle, and ''h'' is the ' ...

equals one-half the product of height and base length. In

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, any two points determine a unique line segment situated within a unique

straight line In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimens ...

, and any three points that do not all lie on the same straight line determine a unique triangle situated within a unique flat plane. More generally, four points in

three-dimensional Euclidean space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-dim ...

determine a

solid figure Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inte ...

called ''

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

''. In

non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

, three "straight" segments (having zero

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

) also determine a "triangle", for instance, a

spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...

hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three point (geometry), points called ''angles'' or ''vertices''. Just as in the Euclidea ...

. A

geodesic triangle In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...

is a region of a general two-dimensional

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

enclosed by three sides that are straight relative to the surface (

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

s). A '' triangle'' is a shape with three

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

d sides, for instance, a '' circular triangle'' with circular-arc sides. (This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted.) Triangles are classified into different types based on their angles and the lengths of their sides. Relations between angles and side lengths are a major focus of

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

. In particular, the sine, cosine, and tangent functions relate side lengths and angles in

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

Definition, terminology, and types

A triangle is a figure consisting of three line segments, each of whose endpoints are connected. This forms a polygon with three sides and three angles. The terminology for categorizing triangles is more than two thousand years old, having been defined in Book One of Euclid's '' Elements''. The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations. Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

, a triangle with two sides having the same length is an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

, and a triangle with three different-length sides is a ''scalene triangle''. A triangle in which one of the angles is a

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

is a

, a triangle in which all of its angles are less than that angle is an

acute triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one ''obtuse angle'' (greater than 90°) and two acute angles. Since a triang ...

, and a triangle in which one of it angles is greater than that angle is an

obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one '' obtuse angle'' (greater than 90°) and two acute angles. Since a trian ...

. These definitions date back at least to

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

. Triangle.Equilateral.svg,

Equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

Triangle.Isosceles.svg,

Isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

Triangle.Scalene.svg, Scalene triangle Triangle.Right.svg,

Right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

Triangle.Acute.svg,

Acute triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one ''obtuse angle'' (greater than 90°) and two acute angles. Since a triang ...

Triangle.Obtuse.svg,

Obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one '' obtuse angle'' (greater than 90°) and two acute angles. Since a trian ...

Appearances

All types of triangles are commonly found in real life. In man-made construction, the isosceles triangles may be found in the shape of

gable A gable is the generally triangular portion of a wall between the edges of intersecting roof pitches. The shape of the gable and how it is detailed depends on the structural system used, which reflects climate, material availability, and aesth ...

s and

pediment Pediments are a form of gable in classical architecture, usually of a triangular shape. Pediments are placed above the horizontal structure of the cornice (an elaborated lintel), or entablature if supported by columns.Summerson, 130 In an ...

s, and the equilateral triangle can be found in the yield sign. The faces of the

Great Pyramid of Giza The Great Pyramid of Giza is the largest Egyptian pyramid. It served as the tomb of pharaoh Khufu, who ruled during the Fourth Dynasty of Egypt, Fourth Dynasty of the Old Kingdom of Egypt, Old Kingdom. Built , over a period of about 26 years ...

are sometimes considered to be equilateral, but more accurate measurements show they are isosceles instead. Other appearances are in

heraldic Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, rank and pedigree. Armory, the best-known branc ...

symbols as in the

flag of Saint Lucia The national flag of Saint Lucia consists of a cerulean blue field charged with a golden triangle in front of a white-edged black isosceles triangle. Adopted in 1967 to replace the British Blue Ensign Defacement (flag), defaced with the arms of ...

and

flag of the Philippines The national flag of the Philippines () is a horizontal List of flags by design#Bicolour, bicolor flag with equal bands of royal blue and Crimson, crimson red, with a white, equilateral chevron at the Glossary of vexillology#Flag elements, hois ...

. Triangles also appear in three-dimensional objects. A

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

is a solid whose boundary is covered by flat

polygonal In geometry, a polygon () is a plane (mathematics), plane Shape, figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its ''edge (geometry), edges'' or ''sides''. The p ...

s known as the faces, sharp corners known as the vertices, and line segments known as the edges. Polyhedra in some cases can be classified, judging from the shape of their faces. For example, when polyhedra have all equilateral triangles as their faces, they are known as

deltahedra A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter Delta (letter), delta resembling a triangular shape Δ. Deltahedra can be categorized by the prope ...

Antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...

s have alternating triangles on their sides.

Pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...

s and

bipyramid In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...

s are polyhedra with polygonal bases and triangles for lateral faces; the triangles are isosceles whenever they are right pyramids and bipyramids. The Kleetope of a polyhedron is a new polyhedron made by replacing each face of the original with a pyramid, and so the faces of a Kleetope will be triangles. More generally, triangles can be found in higher dimensions, as in the generalized notion of triangles known as the

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

, and the

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

s with triangular

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...

s known as the

simplicial polytope In geometry, a simplicial polytope is a polytope whose facet_(mathematics), facets are all Simplex, simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only Triangle, triangular facesCeva'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 In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

; here

Menelaus' theorem In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle , and a Transversal (geometry), transversal line that crosses at points respectively, wi ...

gives a useful general criterion. In this section, just a few of the most commonly encountered constructions are explained. A

perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''se ...

of a side of a triangle is a straight line passing through the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

of the side and being perpendicular to it, forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's

circumcenter 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 circumcen ...

; this point is the center of the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

, the circle passing through all three vertices.

Thales' theorem In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as pa ...

implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. 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 Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...

of a triangle is a straight line through a vertex and perpendicular to 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 The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. An

angle bisector In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''se ...

of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the

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 bis ...

, which is the center of the triangle's

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 ...

. The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the

excircle 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. ...

s; they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form an

orthocentric system In geometry, an orthocentric system is a set (mathematics), set of four point (geometry), points on a plane (mathematics), plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through ...

. 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

. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three

s. The orthocenter (blue point), the center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) 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. Generally, the incircle's center is not located on Euler's line. A

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

of a triangle is a straight line through a vertex and the

of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

or geometric barycenter. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

: the object can be balanced on its centroid in a uniform gravitational field. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. If one reflects a median in 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.

Angles

The sum of the measures of the interior angles of a triangle in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

is always 180 degrees. This fact is equivalent to Euclid's

parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...

. This allows the determination of the measure of the third angle of any triangle, given the measure of two angles. An '' exterior angle'' of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has. Another relation between the internal angles and triangles creates a new concept of

trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

s. The primary trigonometric functions are

sine and cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...

, as well as the other functions. They can be defined as the ratio between any two sides of a right triangle. In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use the

law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...

and the

law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...

. Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A degenerate triangle, whose vertices are

, has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.) The conditions for three angles

\alpha

\beta

, and

\gamma

, each of them between 0° and 180°, to be the angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles

\alpha

\beta

, and

\gamma

exists

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

\cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos(\alpha)\cos(\beta)\cos(\gamma) = 1.

Similarity and congruence

Two triangles are said to be '' similar'', if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Some basic

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

s about similar triangles are: *

If and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar. * If and only if one pair of corresponding sides of two triangles are in the same proportion as another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar. (The ''included angle'' for any two sides of a polygon is the internal angle between those two sides.) * If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar. Two triangles that are

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

have exactly the same size and shape. All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. This is a total of six equalities, but three are often sufficient to prove congruence. Some individually

necessary and sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

s for a pair of triangles to be congruent are: * SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure. * ASA: Two interior angles and the side between them in a triangle have the same measure and length, respectively, as those in the other triangle. (This is the basis of surveying by triangulation.) * SSS: Each side of a triangle has the same length as the corresponding side of the other triangle. * AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as ''AAcorrS'' and then includes ASA above.)

Area

In the Euclidean plane,

area 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-di ...

is defined by comparison with a square of side length , which has area 1. There are several ways to calculate the area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side (the base) times the corresponding altitude :

T = \tfracbh.

This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base and height .

If two sides and and their included angle

\gamma

are known, then the altitude can be calculated using trigonometry, , so the area of the triangle is:

T = \tfracab \sin \gamma.

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...

, named after Heron of Alexandria, is a formula for finding the area of a triangle from the lengths of its sides

a

b

c

. Letting

s = \tfrac12(a + b + c)

be the semiperimeter,

T = \sqrt.

Because the ratios between areas of shapes in the same plane are preserved by

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...

s, the relative areas of triangles in any

affine plane In geometry, an affine plane is a two-dimensional affine space. Definitions There are two ways to formally define affine planes, which are equivalent for affine planes over a field. The first way consists in defining an affine plane as a set on ...

can be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with the same base and oriented area has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's ''Elements''. Given

affine coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

(such as

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

) , , for the vertices of a triangle, its relative oriented area can be calculated using the

shoelace formula The shoelace formula, also known as Gauss's area formula and the surveyor's formula, is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the ...

\begin
T &= \tfrac12 \beginx_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end
= \tfrac12 \begin x_A & x_B \\ y_A & y_B \end
+ \tfrac12 \begin x_B & x_C \\ y_B & y_C \end
+ \tfrac12 \begin x_C & x_A \\ y_C & y_A \end \\
&= \tfrac12(x_Ay_B - x_By_A + x_By_C - x_Cy_B + x_Cy_A - x_Ay_C),
\end

where

,  \cdot ,

is the matrix determinant.

Possible side lengths

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 Degeneracy (mathematics)#T ...

states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. The sum of two side lengths can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices.

Rigidity

Unlike a rectangle, which may collapse into a

from pressure to one of its points, triangles are sturdy because specifying the lengths of all three sides determines the angles. Therefore, a triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. Triangles are strong in terms of rigidity, but while packed in a

tessellating A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensiona ...

arrangement triangles are not as strong as

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

s under compression (hence the prevalence of hexagonal forms in

nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...

). Tessellated triangles still maintain superior strength for

cantilever A cantilever is a rigid structural element that extends horizontally and is unsupported at one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a cantilev ...

ing, however, which is why engineering makes use of tetrahedral trusses.

Triangulation

Triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...

means the partition of any planar object into a collection of triangles. For example, in

polygon triangulation In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is . Triangulations may ...

, a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon. In the case of a

simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...

with sides, there are

n - 2

triangles that are separated by

n - 3

diagonals. Triangulation of a simple polygon has a relationship to the

ear In vertebrates, an ear is the organ that enables hearing and (in mammals) body balance using the vestibular system. In humans, the ear is described as having three parts: the outer ear, the middle ear and the inner ear. The outer ear co ...

, a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears.

Location of a point

One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in the

Cartesian plane In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

, and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane. Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which gives a congruent triangle, or even by rescaling it to a similar triangle: *

Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

specify the relative distances of a point from the sides, so that coordinates

x : y : z

indicate that the ratio of the distance of the point from the first side to its distance from the second side is

x : y

, etc. * Barycentric coordinates of the form

\alpha :\beta :\gamma

specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point.

Related figures

Figures inscribed in a triangle

As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse. This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle

ABC

, let the foci be

P

and

Q

, then:

\frac + \frac + \frac = 1.

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. The intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended). Calabi triangle

The inscribed squares tangent their vertices to the triangle's sides is the special case of

inscribed square problem The inscribed square problem, also known as the square peg problem or the Toeplitz conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is true if the curve is ...

, although the problem asking for a square whose vertices lie on a simple closed curve. A notable example of this figure relation is the

Calabi triangle The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an isosceles triangle which is obtuse triangle, obtuse with an irrati ...

in which the vertices of every three squares are tangent to all obtuse triangle's sides. Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle, two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two ''distinct'' inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has a side of length

q_a

and the triangle has a side of length

a

, part of which side coincides with a side of the square, then

q_a

a

h_a

from the side

a

, and the triangle's area

T

are related according to

q_a=\frac = \frac.

The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when

a^2 = 2T

q = a/2

, and the altitude of the triangle from the base of length

a

is equal to

a

. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is

2\sqrt/3

. Both of these extreme cases occur for the isosceles right triangle.

The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle. Every

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

with area

T

can be inscribed in a triangle of area at most equal to

2T

. Equality holds only if the polygon is a

Figures circumscribed about a triangle

The

tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus ...

of a reference triangle (other than a right triangle) is the triangle whose sides are on the

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

s to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Furthermore, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is unique

conic A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, thou ...

that passes through the triangle's three vertices, its centroid, and its circumcenter. Of all triangles contained in a given

, one with maximal area can be found in linear time; its vertices may be chosen as three of the vertices of the given polygon.

Miscellaneous triangles

Circular triangles

A circular triangle is a triangle with circular arc edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward). The intersection of three disks forms a circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a Reuleaux triangle, which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by the Mohr–Mascheroni theorem. Alternatively, it can be constructed by rounding the sides of an equilateral triangle. A special case of concave circular triangle can be seen in a pseudotriangle. A pseudotriangle is a simply-connected subset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called the ''cusp points''. Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks and bitangent lines, a process known as pseudo-triangulation. For

n

disks in a pseudotriangle, the partition gives

2n - 2

pseudotriangles and

3n - 3

bitangent lines. The

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of any pseudotriangle is a triangle.

Triangle in non-planar space

A non-planar triangle is a triangle not embedded in a

, roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in

hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...

and

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applicati ...

. A triangle in hyperbolic space is called a

, and it can be obtained by drawing on a negatively curved surface, such as a saddle surface. Likewise, a triangle in spherical geometry is called a

, and it can be obtained by drawing on a positively curved surface such as a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

. The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°. In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By Girard's theorem, the sum of the angles of a triangle on a sphere is

180^\circ \times (1 + 4f)

, where

f

is the fraction of the sphere's area enclosed by the triangle. In more general spaces, there are

comparison theorem In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry. Differential e ...

s relating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or elliptic space. For example, a CAT(k) space is characterized by such comparisons.

Fractal geometry

Fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...

shapes based on triangles include the Sierpiński gasket and the

Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...

References

Notes

Footnotes

Works cited

* * * * * * * * * * * * * * * * * * *
Dover reprint, 1956
. * * * * * * * * * * * * * * * * * * * * * * * *

External links

* * Clark Kimberling

Lists some 5200 interesting points associated with any triangle. {{Authority control