"ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal. Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles since no angles are equal. Triangles can also be classified according to"
"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"
"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"""
"Heronian triangle In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense. Any right-angled triangle whose sidelengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the"
"of recognition of the concept of ""betweenness"" and the resulting ambiguity of ""inside"" versus ""outside"" of figures. Isosceles triangle In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having ""exactly"" two sides of equal length, and sometimes as having ""at least"" two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient"
"of the form formula_29, where ""k"" is 7, 97, 1351, 18817, …. The numbers in this sequence have the property that ""k"" consecutive integers have integral standard deviation. Heronian triangle In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense. Any right-angled triangle whose sidelengths are"
"Isosceles triangle In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having ""exactly"" two sides of equal length, and sometimes as having ""at least"" two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and"
"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"
"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"
"within a regular icosahedron of side length ""c"": the shortest line segment from any vertex ""V"" to the plane of its five neighbors has length ""a"", and the endpoints of this line segment together with any of the neighbors of ""V"" form the vertices of a right triangle with sides ""a"", ""b"", and ""c"". 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"
"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"
"Cathetus In a right triangle, a cathetus (originally from the Greek word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a ""side about the right angle"". The side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as ""the other two sides"". If the catheti of a right triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor (shorter) and major (longer)"
"""c"" is the length of the hypotenuse of a right triangle having sides ""a"" and ""b"". For instance, in the illustration to the left, the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple (5,12,13). By overlaying a square grid of side length ""c"" onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides ""a"" and ""b"" into a single square of side ""c"", showing that the"
"the circumcenter is located on a side of the triangle, then the opposite angle is a right one. 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"
"""x"", then If two obtuse triangles have sides (""a, b, c"") and (""p, q, r"") with ""c"" and ""r"" being the respective longest sides, then The Calabi triangle, which is the only non-equilateral triangle for which the largest square that fits in the interior can be positioned in any of three different ways, is obtuse and isosceles with base angles 39.1320261...° and third angle 101.7359477...°. The equilateral triangle, with three 60° angles, is acute. The Morley triangle, formed from any triangle by the intersections of its adjacent angle trisectors, is equilateral and hence acute. The golden triangle is the isosceles"
"its sides be in a geometric progression. The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in an arithmetic progression. Let be the side length of a regular decagon inscribed in the unit circle, where ""φ"" is the golden ratio. Let be the side length of a regular hexagon in the unit circle, and let {2}}</math>}} be the side length of a regular pentagon in the unit circle. Then , so these three lengths form the sides of a right triangle. The same triangle forms half of a golden rectangle. It may also be found"
"of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. Three given angles form a non-degenerate triangle (and indeed an infinitude of them) if and only if both"
"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. There are various standard methods for calculating the length of a side or the measure of an angle. Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations. In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known"
"is 90°, leaving the remaining angle to be 30°. Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees. (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio where ""m"" and ""n"" are any positive integers such that . There are several Pythagorean triples which are well-known, including those with sides in the"
"tetrakis hexahedron, pentakis dodecahedron, and triakis icosahedron, each have isosceles-triangle faces, as do infinitely many pyramids and bipyramids. For any isosceles triangle, the following six line segments coincide: Their common length is the height formula_1 of the triangle. If the triangle has equal sides of length formula_2 and base of length formula_3, the general triangle formulas for the lengths of these segments all simplify to This formula can also be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles. The Euler line of any triangle"
"sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides ""a"", ""b"", ""c"" can"
"their internal angles, measured here in degrees. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a ""2-simplex"" (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4"
"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"
"with angles α, β and γ is similar to a triangle with side lengths equal to sin α, sin β and sin γ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sin α, sin β and sin γ. The side whose length is sin α is opposite to the angle whose measure is α, etc. The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the"
"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"
"Rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is a parallelogram and"
"Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse"
"two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the ""vertex angle"" and the angles that have the base as one of their sides are called the ""base angles"". The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle"
"a triangle with any two sides ""a"" and ""b"", and an angle ""C"" between them is Each angle of an equilateral triangle is 60°, so The sine of 60° is formula_50. Thus since all sides of an equilateral triangle are equal. Equilateral triangles have frequently appeared in man made constructions: 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,"
"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"
"Sierpinski triangle The Sierpinski triangle (also with the original orthography ""Sierpiński""), also called the Sierpinski gasket or the Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e., it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński. There are many different ways of constructing"