Triangle
A triangle holds the distinction of being one of the basic shapes in geometry, built from just three corners and three sides. The corners, called vertices, are zero-dimensional points. The sides connecting them, called edges, are one-dimensional line segments. From this minimal arrangement comes a figure whose three internal angles always sum to a straight angle, 180 degrees or pi radians. The terminology for sorting these shapes is more than two thousand years old, set down in Book One of Euclid's Elements. So how does a figure this simple generate centers, circles, and lines named after mathematicians across centuries? Why do engineers reach for it when they want something that will not collapse? And what happens to its tidy rules when you draw it on a sphere or a saddle instead of a flat page?
Euclid's Greek terms, transliterated directly or translated into Latin, still govern how triangles are named today. A triangle whose sides are all the same length is equilateral. One with two sides of equal length is isosceles, and one with three different lengths is scalene. The angles supply a second naming system that runs in parallel. A triangle containing a right angle is a right triangle. When every angle stays below that, the triangle is acute, and when one angle exceeds it, the triangle is obtuse. These definitions reach back at least to Euclid himself, who first separated the categories in Book One of his Elements.
The yield sign carries an equilateral triangle that any driver would recognize. Gables and pediments in man-made construction take the form of isosceles triangles, and the flag of Saint Lucia and the flag of the Philippines both place the shape into heraldic symbols. The faces of the Great Pyramid of Giza are sometimes called equilateral, though more accurate measurements show them to be isosceles instead. Triangles climb into three dimensions as well, covering the faces of solids called polyhedra. When a polyhedron has all equilateral triangles for its faces, it is known as a deltahedron. Antiprisms carry alternating triangles along their sides, while pyramids and bipyramids use triangles for their lateral faces. The Kleetope replaces each face of a polyhedron with a pyramid, so its faces become triangles too. Push into higher dimensions and the triangle generalizes into the simplex, the seed of the simplicial polytopes.
Ceva's theorem gives a criterion for deciding when three lines drawn through a triangle meet at a single point, and that idea unlocks a crowd of special points hiding inside the figure. The three perpendicular bisectors of the sides converge at the circumcenter, the center of the circumcircle that passes through all three vertices. Where the circumcenter sits reveals the triangle's character. Inside means acute, outside means obtuse, and resting on a side means a right angle, by Thales' theorem. The three altitudes, each running from a vertex perpendicular to the opposite side, meet at the orthocenter, which lies inside the triangle exactly when the triangle is acute. The three angle bisectors converge at the incenter, the center of the incircle, the largest circle that fits inside and touches all three sides. Three excircles lie outside, each touching one side and the extensions of the other two. The midpoints of the sides and the feet of the altitudes share a single circle, the nine-point circle, whose radius is half that of the circumcircle. The orthocenter, the center of the nine-point circle, the centroid, and the circumcenter all sit on one line called Euler's line. The three medians, each joining a vertex to the midpoint of the opposite side, meet at the centroid, which cuts every median in a 2-1 ratio. Cut a triangle from a thin sheet of uniform density and it balances on that very point.
The 180-degree rule for a triangle's interior angles is equivalent to Euclid's parallel postulate, so the two stand or fall together. Knowing two angles immediately fixes the third. An exterior angle, supplementary to its interior partner, equals the sum of the two interior angles not adjacent to it, a result called the exterior angle theorem. Add up the three exterior angles and you always reach 360 degrees, a total that holds for any convex polygon regardless of how many sides it has. Specifying the three angles does not pin down the size, so infinitely many triangles can share the same set of angles. A degenerate triangle, whose vertices fall on a single line, carries internal angles of 0 degrees and 180 degrees, and whether it counts as a triangle at all is a matter of convention. The ratios between sides of a right triangle give rise to the sine and cosine functions, the tools that let the law of sines and the law of cosines recover an unknown side or angle in a scalene triangle.
Two triangles are similar when every angle of one matches the corresponding angle of the other, which forces their sides into the same proportion. Matching just two pairs of angles is enough to prove it. So is a single pair of sides in proportion with their included angles equal, or all three pairs of sides sharing one proportion. Congruence is the stronger relation, demanding the same size and shape, a total of six matching equalities between angles and sides. Three of those equalities usually suffice. The SAS postulate pairs two sides and the included angle, while SSS compares all three sides directly. ASA matches two angles and the side between them, the principle behind surveying by triangulation, and AAS uses two angles with a non-included side.
Area in the Euclidean plane is measured against a square of side length one, which is assigned an area of 1. The oldest and simplest method for a triangle takes half the product of a base and its corresponding altitude, a result proven by cutting a triangle and its copy into pieces that rearrange into a rectangle. Heron's formula, named after Heron of Alexandria, reaches the same area from the three side lengths alone by way of the semiperimeter. Because affine transformations preserve the ratios between areas, relative areas can be defined without any notion of distance, an approach laid out in Book One of Euclid's Elements. The shoelace formula computes a triangle's oriented area straight from the Cartesian coordinates of its vertices using a matrix determinant. The triangle inequality sets the boundary on which side lengths are even possible. The sum of any two sides must be greater than or equal to the third, with equality only in the degenerate case of collinear vertices.
Press on a corner of a rectangle and it folds into a parallelogram, but a triangle refuses. Fixing the lengths of all three sides locks the angles in place, so each side supports the other two and the shape holds unless something bends, stretches, or breaks. This is why structural quadrilaterals are braced with a diagonal that splits them into two rigid triangles. Packed in a tessellation, triangles yield to hexagons under compression, which is part of why hexagonal forms appear so often in nature. Yet triangles keep the upper hand for cantilevering, the reason engineering relies on tetrahedral trusses. The flat-page rules loosen entirely off the plane. On a negatively curved saddle surface, a hyperbolic triangle's angles sum to less than 180 degrees, while on a sphere a spherical triangle exceeds it. A triangle can be drawn on a sphere with three right angles, each 90 degrees, adding up to 270. By Girard's theorem, the angle excess of a spherical triangle measures the fraction of the sphere's area it encloses. The same three-sided form even seeds fractals, repeating into the Sierpinski gasket and the Koch snowflake.
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Common questions
What is a triangle in geometry?
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. Its corners are called vertices and its sides are called edges, and its three internal angles always sum to a straight angle of 180 degrees or pi radians.
What are the types of triangles by sides and angles?
By side length, a triangle is equilateral when all three sides are equal, isosceles when two sides are equal, and scalene when all three differ. By angle, it is a right triangle when one angle is a right angle, acute when all angles are smaller, and obtuse when one angle is larger. These definitions date back at least to Euclid.
Why are triangles used in engineering and construction?
Triangles are rigid because specifying the lengths of all three sides determines the angles, so the shape will not collapse unless its sides bend, stretch, or break. Structural quadrilaterals are often braced with a diagonal that splits them into two rigid triangles, and engineering uses tetrahedral trusses for cantilevering.
What are the main points and circles associated with a triangle?
A triangle has a circumcenter at the center of the circumcircle, an incenter at the center of the incircle, an orthocenter where its altitudes meet, and a centroid where its medians meet. The orthocenter, the nine-point circle's center, the centroid, and the circumcenter all lie on Euler's line.
Do the angles of a triangle always add up to 180 degrees?
In Euclidean space the interior angles of a triangle always add up to 180 degrees, a fact equivalent to Euclid's parallel postulate. On a negatively curved surface a hyperbolic triangle sums to less than 180 degrees, and on a sphere a spherical triangle sums to more than 180 degrees.
How do you calculate the area of a triangle?
One of the oldest methods takes half the product of a base and its corresponding altitude. Heron's formula, named after Heron of Alexandria, finds the area from the three side lengths using the semiperimeter, and the shoelace formula computes it from the Cartesian coordinates of the vertices.
All sources
66 references cited across the entry
- 7BookGeometry Turned On: Dynamic Software in Learning, Teaching, and ResearchSchattschneider, Doris et al. — The Mathematical Association of America — 1997
- 8JournalOrthocentric simplices and biregularityAllan L. Edmonds et al. — 2008
- 9JournalMetric spaces in which all triangles are degenerateBettina Richmond et al. — 1997
- 10ThesisMaking sense of definitions in geometry: Metric-combinatorial approaches to classifying triangles and quadrilateralsOrlando Braulio Alonso — Teachers College, Columbia University — 2009
- 11JournalThe Surveyor's Area FormulaBart Braden — 1986
- 13JournalTwenty-one points on the nine-point circleClark Kimberling — March 2008
- 14JournalReflection-Induced Perspectivities Among TrianglesPeter Moses et al. — 2009
- 15JournalEuler and triangle geometryGeoff Smith et al. — November 2007
- 16JournalExtremal area ellipses of a convex quadrilateralJohn R. Silvester — March 2017
- 17JournalThe Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the TriangleR. H. Eddy et al. — 1994
- 19The area of a spherical triangle. Girard's Theorem.John C. Polking — 1999-04-25
- 21BookFractal Worlds: Grown, Built, and ImaginedMichael Frame et al. — Yale University Press — 2016-06-21
- 22JournalProving a nineteenth century ellipse identity"Patricia R. Allaire et al. — 2012
- 23BookThe Heritage of ThalesW. S. Anglin et al. — Springer — 1995
- 24BookLinear AlgebraTom M. Apostol — Wiley — 1997
- 25BookAlgebra and TrigonometrySheldon Axler — John Wiley & Sons — 2012
- 26JournalSquares inscribed in angles and trianglesHerbert Bailey et al. — 1998
- 27BookLectures on spaces of nonpositive curvatureWerner Ballmann — Birkhäuser Verlag — 1995
- 28BookA panoramic view of Riemannian geometryMarcel Berger — Springer — 2002
- 29BookComputational geometry: algorithms and applicationsMark Theodoor de Berg et al. — Springer — 2000
- 30BookThe First Six Books of the Elements of EuclidOliver Byrne — TASCHEN GmbH — 2013
- 31BookPolyhedraPeter R. Cromwell — Cambridge University Press — 1997
- 32JournalDeltahedraH. Martyn Cundy — 1952
- 33BookProblems in Euclidean Space: Applications of ConvexityH. G. Eggleston — Dover Publications — 2007
- 34JournalA parallel algorithm for enclosed and enclosing trianglesSharat Chandran et al. — 1992
- 35BookGeometry RevisitedH. S. M. Coxeter et al. — Mathematical Association of America — 1967
- 36BookDiscrete and Computational GeometrySatyan L. Devadoss et al. — Princeton University Press — 2011
- 37BookReal-Time Collision DetectionChrister Ericson — CRC Press — 2005
- 38BookHistorical Dictionary of the PhilippinesArtemio R. Guillermo — Scarecrow Press — 2012
- 39BookThe Cartoon Guide to GeometryLarry Gonick — William Morrow — 2024
- 40BookStructure and Form in Design: Critical Ideas for Creative PracticeMichael Hann — A&C Black — 2014
- 41BookThe Thirteen Books of Euclid's ElementsThomas L. Heath — Cambridge University Press — 1926
- 42BookThe Shape of the Great PyramidRoger Herz-Fischler — Wilfrid Laurier University Press — 2000
- 43BookGeometry: Our Cultural HeritageA. Holme — Springer — 2010
- 44JournalA short elementary proof of the Mohr-Mascheroni theoremNorbert Hungerbühler — 1994
- 45BookMathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical SciencesD. W. Jordan et al. — Oxford University Press — 2010
- 46JournalAn Elementary Proof of Marden's TheoremDan Kalman — 2008
- 47BookGeometry Transformed: Euclidean Plane Geometry Based on Rigid MotionsJames R. King — American Mathematical Society — 2021
- 48BookGeometry: A High School CourseSerge Lang et al. — Springer — 1988
- 49BookA Treatise on Geometry and Its Application in the ArtsDionysius Lardner — 1840
- 50JournalOn the ratio of the inradius to the circumradius of a triangleMichael S. Longuet-Higgins — 2003
- 51JournalPolygons have earsG. H. Meisters — 1975
- 52BookOrigami Polyhedra DesignJohn Montroll — A K Peters — 2009
- 53BookProgress in Information Geometry: Theory and ApplicationsFrank Nielsen — Springer — 2021
- 54JournalComputer Aided Research into Triangle GeometryAdrian Oldknow — 1995
- 55JournalWhy Are the Side Lengths of the Squares Inscribed in a Triangle so Close to Each Other?Victor Oxman et al. — 2013
- 56BookGeneral Chemistry: Principles and Modern ApplicationsR. H. Petrucci et al. — Prentice Hall — 2002
- 57BookAdvances in Discrete and Computational Geometry: Proceedings of the 1996 AMS-IMS-SIAM Joint Summer Research Conference, Discrete and Computational Geometry—Ten Years Later, July 14-18, 1996, Mount Holyoke CollegeMichel Pocchiola et al. — American Mathematical Soc. — 1999
- 58BookIntroduction to Hyperbolic GeometryArlan Ramsay et al. — Springer — 1995
- 59JournalAn Investigation of Preferred Shapes for Warning LabelsMichael W. Riley et al. — December 1982
- 60BookGeometry For DummiesMark Ryan — John Wiley & Sons — 2008
- 61BookMethods of GeometryJames T. Smith — Wiley — 2000
- 62BookGeometry from Euclid to KnotsSaul Stahl — Prentice-Hall — 2003
- 63BookThe Classification of Quadrilaterals: A Study in DefinitionZalman Usiskin et al. — Information Age Publishing — 2008
- 64BookAlgorithmic Foundation of Robotics VII: Selected Contributions of the Seventh International Workshop on the Algorithmic Foundations of RoboticsMostafa Vahedi et al. — 2008
- 65JournalSimple trigonometric substitutions with broad resultsVardan Verdiyan et al. — 2007
- 66BookTrigonometryCynthia Young — John Wiley & Sons — 2017