"Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. 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. Denoting the common length of the sides of the equilateral triangle as ""a"", we can determine using the Pythagorean theorem that: Denoting the radius of the circumscribed circle as ""R"", we can determine using trigonometry that: Many of these quantities"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"end at an ideal vertex represented as an ""omega point"". Such a pair of sides may also be said to form an angle of zero. A triangle with a zero angle is impossible in Euclidean geometry for straight sides lying on distinct lines. However, such zero angles are possible with tangent circles. A triangle with one ideal vertex is called an omega triangle. Special Triangles with ideal vertices are: A triangle where one vertex is an ideal point, one angle is right: the third angle is the angle of parallelism for the length of the side between the right and"
"Pons asinorum In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (; ), typically translated as ""bridge of asses"". This statement is Proposition 5 of Book 1 in Euclid's ""Elements"", and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The name of this statement is also used metaphorically for a problem or challenge which will separate the sure of mind from the simple, the"
"was also known to be just one example of a general rule that any triangle where the length of two sides, each squared and then added together , equals the square of the length of the third side , would also be a right angle triangle. This is now known as the Pythagorean theorem, and a triple of numbers that meets this condition is called a Pythagorean triple – both are named after the ancient Greek Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been"
"the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another."" Its name may be attributed to its frequent role as the first real test in the ""Elements"" of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them"
"A triangle (""X"", ""Y"", ""Z"", ""u"", ""v"", ""w"") consists of 3 objects ""X"", ""Y"", and ""Z"", together with morphisms ""u"" : ""X"" → ""Y"", ""v"" : ""Y"" → ""Z"" and ""w"" : ""Z"" → ""X""[1]. Triangles are generally written in the unravelled form: Triangulated category In mathematics, a triangulated category is a category together with the additional structure of a ""translation functor"" and a class of ""distinguished triangles"". Prominent examples are the derived category of an abelian category and the stable homotopy category of spectra (more generally, the homotopy category of a stable ∞-category), both of which carry the structure"
"Trigonal planar molecular geometry In chemistry, trigonal planar is a molecular geometry model with one atom at the center and three atoms at the corners of an equilateral triangle, called peripheral atoms, all in one plane. In an ideal trigonal planar species, all three ligands are identical and all bond angles are 120°. Such species belong to the point group D. Molecules where the three ligands are not identical, such as HCO, deviate from this idealized geometry. Examples of molecules with trigonal planar geometry include boron trifluoride (BF), formaldehyde (HCO), phosgene (COCl), and sulfur trioxide (SO). Some ions with trigonal"
"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. 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 theorems about similar triangles are: Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles"
"the third angle. The triangle where two vertices are ideal points and the remaining angle is right, one of the first hyperbolic triangles (1818) described by Ferdinand Karl Schweikart. The triangle where all vertices are ideal points, an ideal triangle is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles. The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles. The length"
"Sum of angles of a triangle In several geometries, a triangle has three ""vertices"" and three ""sides"", where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, radians, two right angles, or a half-turn. It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the"
"Triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ""A"", ""B"", and ""C"" is denoted formula_1. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces,"
"Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ""oblique"". A uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces"
"angle A"" and ""opposed to angle B"". If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area ""T"" is where ""a"""
"Trirectangular tetrahedron In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the ""right angle"" of the trirectangular tetrahedron and the face opposite it is called the ""base"". The three edges that meet at the right angle are called the ""legs"" and the perpendicular from the right angle to the base is called the ""altitude"" of the tetrahedron. Only the bifurcating graph of the formula_1 Affine Coxeter group has a Trirectangular tetrahedron fundamental domain. If the legs have lengths ""a, b, c"", then the trirectangular tetrahedron has"
"are the same triangle. A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t{2,3}. Alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product {3}x{}. The dual of a triangular prism is a triangular bipyramid. The symmetry group of a right 3-sided prism with triangular base is ""D"" of order 12. The rotation group is ""D"" of order 6."
"appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends"
"Tri-oval A tri-oval is a shape which derives its name from the two other shapes it most resembles, a triangle and an oval. Rather than meeting at sharp, definable angles as the sides of a triangle do, in a tri-oval these angles are instead rounded into smooth curves. While an oval has four turns, a tri-oval has six. More formally, according to the four-vertex theorem, every smooth simple closed curve has at least four vertices, points where its curvature reaches a local minimum or maximum. In a tri-oval, there are six such points, alternating between three minima and three maxima."
"is the only triangle with integer sides and three rational angles as measured in degrees. The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one). Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral"
"Golden triangle (mathematics) A golden triangle, also known as the sublime triangle,is an isosceles triangle in which the duplicated side is in the golden ratio to the distinct side: Golden triangles are found in the nets of several stellations of dodecahedrons and icosahedrons. Also, it is the shape of the triangles found in the points of pentagrams. The vertex angle is equal to Since the angles of a triangle sum to 180°, base angles are therefore 72° each. The golden triangle can also be found in a regular decagon, or an equiangular and equilateral ten-sided polygon, by connecting any two"
"Heptagonal triangle A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter and longer diagonals of the regular heptagon. All heptagonal triangles are similar (have the same shape), and so they are collectively known as ""the"" heptagonal triangle. Its angles have measures formula_1 and formula_2 and it is the only triangle with angles in the ratios 1:2:4. The heptagonal triangle has various remarkable properties. The heptagonal triangle's nine-point center is also its"
"Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two Latin words ""quadri"", a variant of four, and ""latus"", meaning ""side"". Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ""ABCD"" add up to 360 degrees of arc, that is"
"with sides ""a"", ""b"" and ""c"", and every such triangle has a right angle between the sides of lengths ""a"" and ""b"". An alternative statement is: For any triangle with sides ""a"", ""b"", ""c"", if then the angle between ""a"" and ""b"" measures 90°. This converse also appears in Euclid's ""Elements"" (Book I, Proposition 48): It can be proven using the law of cosines or as follows: Let ""ABC"" be a triangle with side lengths ""a"", ""b"", and ""c"", with Construct a second triangle with sides of length ""a"" and ""b"" containing a right angle. By the Pythagorean theorem, it"
"type of triangles: Triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}. Conway calls it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of"
"this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of ""ticks"", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3"
"Special right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an ""angle-based"" right triangle. A ""side-based"" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles"
"not) satisfy formula_151. We can generate the full equivalence class of similar triangles that satisfy B=3A by using the formulas where formula_28 and formula_27 are integers such that formula_157. Note that all triangles with B = 3A (whether with integer sides or not) satisfy formula_158. The only integer triangle with three rational angles (rational numbers of degrees, or equivalently rational fractions of a full turn) is the equilateral triangle. This is because integer sides imply three rational cosines by the law of cosines, and by Niven's theorem a rational cosine coincides with a rational angle if and only if the"
"Ideal triangle In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ""triply asymptotic triangles"" or ""trebly asymptotic triangles"". The vertices are sometimes called ideal vertices. All ideal triangles are congruent. Ideal triangles have the following properties: In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties: formula_1 formula_2 . If the curvature is −""K"" everywhere rather than −1, the areas above should be multiplied by 1/""K"" and the lengths and distances should be multiplied by 1/."
"or formula_16 with coprime formula_11. Trirectangular tetrahedron In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the ""right angle"" of the trirectangular tetrahedron and the face opposite it is called the ""base"". The three edges that meet at the right angle are called the ""legs"" and the perpendicular from the right angle to the base is called the ""altitude"" of the tetrahedron. Only the bifurcating graph of the formula_1 Affine Coxeter group has a Trirectangular tetrahedron fundamental domain. If the legs have lengths ""a, b, c"","
"triangles would be a completely flat figure, and a heptagonal or higher would have the triangles not meet at all. Right pyramids with regular star polygon bases are called star pyramids. For example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ( )∨P, where ( ) is the apex point, ∨ is a join operator, and P is a base polygon. An isosceles triangle right tetrahedron can be written as ( )∨[( )∨{ }] as the join of a point to an isosceles triangle base, as [( )∨( )]∨{"
"functions are true for all angles and are known as ""trigonometric identities"". Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below. In the following identities, ""A"", ""B"" and ""C"" are the angles of a triangle and ""a"", ""b"" and ""c"" are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram). The law of sines (also known as the ""sine rule"") for an arbitrary triangle states: where"
"triangle in which the ratio of the duplicated side to the base side equals the golden ratio. It is acute, with angles 36°, 72°, and 72°, making it the only triangle with angles in the proportions 1:2:2. The heptagonal triangle, with sides coinciding with a side, the shorter diagonal, and the longer diagonal of a regular heptagon, is obtuse, with angles formula_28 and formula_29 The only triangle with consecutive integers for an altitude and the sides is acute, having sides (13,14,15) and altitude from side 14 equal to 12. The smallest-perimeter triangle with integer sides in arithmetic progression, and the"