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Dictionary:

triangle

  (trī'ăng'gəl) pronunciation
triangle
(Click to enlarge)
triangle
top: right triangle
bottom: equilateral triangle
(Academy Artworks)
n.
    1. The plane figure formed by connecting three points not in a straight line by straight line segments; a three-sided polygon.
    2. Something shaped like such a figure: a triangle of land.
  2. Any of various flat, three-sided drawing and drafting guides, used especially to draw straight lines at specific angles.
  3. Music. A percussion instrument consisting of a piece of metal in the shape of a triangle open at one angle.
  4. A relationship involving three people, especially a ménage à trois.

[Middle English, from Old French, from Latin triangulum, from neuter of triangulus, three-angled : tri-, tri- + angulus, angle.]


 
 
Computer Desktop Encyclopedia: triangle

A three-sided polygon. In 3D graphics, the surfaces of 3D objects are broken down into triangles. Small numbers of triangles are used for flat surfaces, while large numbers are used to mold curved surfaces similar to the way a geodesic dome is constructed. The three points of every triangle (vertices) are computed on an X-Y-Z scale and must be recomputed each time the object is moved. See triangle setup and geometry calculations.

Triangles
The smaller the triangle, the more realistic the curve. It takes an enormous number of triangles to simulate totally round objects.


 
Investment Dictionary: Triangle

A technical analysis pattern created by drawing trendlines along a price range that gets narrower over time because of lower tops and higher bottoms. Variations of a triangle include ascending and descending triangles. Triangles are very similar to wedges and pennants.

Investopedia Says:
Technical analysts see a breakout of this triangular pattern as either bullish (on a breakout above the upper line) or bearish (on a breakout below the lower line).

Related Links:
Here we pay some attention to the triangle, usually one of the first chart patterns that a novice technician learns. Continuation Patterns - Part 1
Take a closer look at triangles, which appear in ascending, descending and symmetrical forms Continuation Patterns - Part 2
Take a closer look at Ascending and Descending Triangles. Continuation Patterns - Part 3
Learn how to read these formations of horizontal trading patterns. Triangles: A Short Study in Continuation Patterns
From picking the right type of stock to setting stop-losses, learn how to trade wisely. Day Trading Strategies For Beginners
Learn how chartists analyze the price movements of the market. We'll introduce you to the most important concepts in this approach. Basics Of Technical Analysis


 
Britannica Concise Encyclopedia: triangle

Geometric figure with three sides and three angles. Each two sides meet at a point called a vertex, and the three angles sum to 180°. A triangle with one 90° (right) angle is a right triangle. A triangle with all sides (and thus all angles) equal is equilateral, one with two sides equal is isosceles, and one with no two sides equal is scalene. Triangles are particularly useful in surveying, astronomy, and navigation. Two observation points (sight lines) form a triangle with a reference object serving as one vertex and the observation points as the other two. Knowing the angles of the sight lines and the distance between the observation points allows the calculation of the lengths of the other sides using the methods of trigonometry.

For more information on triangle, visit Britannica.com.

 
Columbia Encyclopedia: triangle,
in mathematics, plane figure bounded by three straight lines, the sides, which intersect at three points called the vertices. Any one of the sides may be considered the base of the triangle. The perpendicular distance from a base to the opposite vertex is called an altitude. The area of a triangle is equal to one half the product of the base and the corresponding altitude. The line segment joining the midpoint of a side to the opposite vertex is called a median. All three altitudes of a triangle go through a single point, and all three medians go through a single (usually different) point. In Euclidean geometry the sum of the angles of a triangle is equal to two right angles (180°). If all three angles of a triangle are equal, the triangle is called equilateral. An isosceles triangle has two equal angles. A scalene triangle is one in which all three angles are different. A right triangle has one right angle. In geometry it is shown that two triangles are congruent (i.e., are the same shape and size) if, in general, any three independent parts (sides or angles) of one are the same as the corresponding three parts of the other. The rules of congruency make it possible, in trigonometry, to compute the sides and the angles of a triangle when three of these values are known. The triangle is the simplest of the polygons (i.e., it has the least possible number of sides). Since any polygon can be broken up into triangles by drawing various diagonals, a complete theory of the measurement of triangles provides a complete theory of the measurement of all polygons. In non-Euclidean geometries, the angles of a triangle are either less than two right angles (hyperbolic geometry) or more than two right angles (elliptic geometry).

 
Veterinary Dictionary: triangle

A three-cornered object, figure or area, as such an area on the surface of the body capable of fairly precise definition. Called also trigone.

  • facial t. — a triangular area whose points are the basion and the alveolar and nasal points.
  • femoral t. — the triangle bounded cranially by the sartorius, caudally by the pectineus and deeply by the iliopsoas muscles in the dog. The pulse of the femoral artery can be taken at this site.
  • vesical t. — the area of the bladder wall within the triangle demarcated by the ureteral and urethral orifices. The bladder mucosa is firmly attached at this point and does not form folds.
  • Viborg's t. — a surgical site on the side of the throat of the horse bounded by the caudal border of the mandible, the linguofacial vein and the tendon of the sternocephalic muscle.
 
Marine Corps Dictionary: Triangle

The civilian community outside MCB Quantico, VA.

 
Word Tutor: triangle
pronunciation

IN BRIEF: A three-sided shape with three corners.

pronunciation She drew a triangle, and another one on top of it, and another one... until they formed a star.

 
Wikipedia: triangle
A triangle.
Enlarge
A triangle.

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are straight line segments.

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

Types of triangles

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

  • In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon[1]
  • In an isosceles triangle, two sides are of equal length. An isosceles triangle also has two congruent angles (namely, the angles opposite the congruent sides). An equilateral triangle is an isosceles triangle, but not all isosceles triangles are equilateral triangles.[2]
  • In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.[3]
Equilateral Triangle Isosceles triangle Scalene triangle
Equilateral Isosceles Scalene

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

  • A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
  • An obtuse triangle has one internal angle larger than 90° (an obtuse angle).
  • An acute triangle has internal angles that are all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.
  • An oblique triangle has only angles that are smaller or larger than 90°. It is therefore any triangle that is not a right triangle.
Right triangle Obtuse triangle Acute triangle
Right Obtuse Acute
  \underbrace{\qquad \qquad \qquad \qquad \qquad \qquad}_{}
  Oblique

Basic facts

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.

The angles of a triangle add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the triangle inequality.

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

A few basic postulates and theorems about similar triangles: Two triangles are similar if at least 2 corresponding angles are congruent. If two corresponding sides of two triangles are in proportion, and their included angles are congruent, the triangles are similar. If three sides of two triangles are in proportion, the triangles are similar.

For two triangles to be congruent, each of their corresponding angles and sides must be congruent (6 total). A few basic postulates and theorems about congruent triangles: SAS Postulate: If two sides and the included angles of two triangles are correspondingly congruent, the two triangles are congruent. SSS Postulate: If every side of two triangles are correspondingly congruent, the triangles are congruent. ASA Postulate: If two angles and the included sides of two triangles are correspondingly congruent, the two triangles are congruent. AAS Theorem: If two angles and any side of two triangles are correspondingly congruent, the two triangles are congruent. Hypotenuse-Leg Theorem: If the hypotenuses and 1 pair of legs of two right triangles are correspondingly congruent, the triangles are congruent.

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

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

The Pythagorean theorem
Enlarge
The Pythagorean theorem

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

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

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

Some other facts about right triangles:

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

For all triangles, angles and sides are related by the law of cosines and law of sines.

Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.

The circumcenter is the center of a circle passing through the three vertices of the triangle.
The circumcenter is the center of a circle passing through the three vertices of the triangle.

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above.

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

The intersection of the altitudes is the orthocenter.
The intersection of the altitudes is the orthocenter.

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

The intersection of the angle bisectors finds the center of the incircle.
The intersection of the angle bisectors finds the center of the incircle.

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


The barycenter is the center of gravity.
The barycenter is the center of gravity.

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

Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.
Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.

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


Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).

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

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

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

Computing the Area (S) of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. The most common and simplest formula is

S=\frac{1}{2}bh

where S is area, b is the length of the base of the triangle, and h is the height or altitude of the triangle. Other approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.[4]

Using vectors

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

The area of triangle ABC is half of this, or S = ½|AB × AC|.

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

\frac{1}{2} \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\frac{1}{2} \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2} \, .
Applying trigonometry to find the altitude h.
Applying trigonometry to find the altitude h.

Using trigonometry

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

S = \frac{1}{2}ab\sin \gamma = \frac{1}{2}bc\sin \alpha = \frac{1}{2}ca\sin \beta.

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

S = \frac{1}{2}ab\sin (\alpha+\beta) = \frac{1}{2}bc\sin (\beta+\gamma) = \frac{1}{2}ca\sin (\gamma+\alpha).

Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area S can be computed as ½ times the absolute value of the determinant

S=\frac{1}{2}\left|\det\begin{pmatrix}x_B & x_C \\ y_B & y_C \end{pmatrix}\right| = \frac{1}{2}|x_B y_C - x_C y_B|.

For three general vertices, the equation is:

S=\frac{1}{2} \left| \det\begin{pmatrix}x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end{pmatrix} \right| = \frac{1}{2} \big| x_A y_C - x_A y_B + x_B y_A - x_B y_C + x_C y_B - x_C y_A \big|.

In three dimensions, the area of a general triangle {A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC)} is the 'Pythagorean' sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0):

S=\frac{1}{2} \sqrt{ \left( \det\begin{pmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} y_A & y_B & y_C \\ z_A & z_B & z_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} z_A & z_B & z_C \\ x_A & x_B & x_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 }.

Using Heron's formula

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

S = \sqrt{s(s-a)(s-b)(s-c)}

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

An equivalent way of writing Heron's formula is

S = \frac{1}{4} \sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}.

Non-planar triangles

A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.

While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°.

See also

References

  1. ^ Eric W. Weisstein, Equilateral triangle at MathWorld.
  2. ^ Eric W. Weisstein, Isosceles triangle at MathWorld.
  3. ^ Eric W. Weisstein, Scalene triangle at MathWorld.
  4. ^ Eric W. Weisstein, Triangle area at MathWorld.

External links

Polygons
Listed by number of sides
1-10 sides Henagon · Digon · Triangle · Quadrilateral · Pentagon · Hexagon  · Heptagon  · Octagon  · Enneagon  · Decagon
11-21 sides Hendecagon  · Dodecagon  · Triskaidecagon  · Tetradecagon  · Pentadecagon  · Hexadecagon  · Heptadecagon  · Octadecagon  · Enneadecagon
Other Chiliagon  · Myriagon

zh-classical:三角形new:त्रिकोणvls:Drieoek


This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

 
Translations: Translations for: Triangle

Dansk (Danish)
n. - trekant, triangel

idioms:

  • isosceles triangle    ligebenet trekant

Nederlands (Dutch)
driehoek, triangel, tekendriehoek

Français (French)
n. - (Math, Mus) triangle

idioms:

  • isosceles triangle    triangle isocèle

Deutsch (German)
n. - Dreieck, Triangel, Zeichenwinkel

idioms:

  • isosceles triangle    gleichschenkliges Dreieck

Ελληνική (Greek)
n. - τρίγωνο

idioms:

  • isosceles triangle    ισοσκελές τρίγωνο

Italiano (Italian)
triangolo, triangolo isoscele, squadra

Português (Portuguese)
n. - triângulo (m)

idioms:

  • isosceles triangle    triângulo isósceles (m)

Русский (Russian)
треугольник

idioms:

  • isosceles triangle    равносторонний треугольник

Español (Spanish)
n. - triángulo, escuadra, cartabón

idioms:

  • isosceles triangle    triángulo isósceles

Svenska (Swedish)
n. - triangel

中文（简体） (Chinese (Simplified))
三角, 三角板, 三角尺, 三角形之物, 三角铁

idioms:

  • isosceles triangle    等腰三角形

中文（繁體） (Chinese (Traditional))
n. - 三角, 三角板, 三角尺, 三角形之物, 三角鐵

idioms:

  • isosceles triangle    等腰三角形

한국어 (Korean)
n. - 삼각형, 트라이앵글, 3각 기중기

日本語 (Japanese)
n. - 三角形, 三角形の物, 三角定規, 3つ組, トライアングル, 三角関係

idioms:

  • right triangle    直角三角形

العربيه (Arabic)
‏(الاسم) ألمثلث اله من ألات ألنقر الموسيقيه قوامها قضيب من فولاذ على شكل مثلث, مثلث‏

עברית (Hebrew)
n. - ‮משולש, מתקן משלוש קורות להעלאת מטען לאוניה‬

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Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved.  Read more
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