Concept
Triangle
Overview
A triangle or trigon is a polygon consisting of three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent sides; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. 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 equals one-half the product of height and base length.
In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, 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 determine a solid figure called tetrahedron.
In non-Euclidean geometries, three "straight" segments also determine a "triangle", for instance, a spherical triangle or hyperbolic triangle. A geodesic triangle is a region of a general two-dimensional surface enclosed by three sides that are straight relative to the surface (geodesics). A curvilinear triangle is a shape with three curved 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. In particular, the sine, cosine, and tangent functions relate side lengths and angles in right triangles.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Triangle is discussed.
When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles; and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles; and out of the…
Plato, Timaeus
names the idea; states what it is; opens on it
In general, we shall take, instead of straight lines, the lines that give the shortest route on the surface from place to place. Such lines are called “geodesics.” On the earth, the geodesics are great circles. In general, they are the shortest way of traveling from point to point if you are unable to leave the surface. They take the place of straight lines in the intrinsic geometry of a surface. When we inquire…
Bertrand Russell, The A B C of Relativity
states what it is; names the idea; opens on it
Wherefore we make one and the same affirmation concerning God when we say that He has from eternity decreed that three angles of a triangle are equal to two right angles, as when we say that He has understood it.
Benedictus de Spinoza, The Philosophy of Spinoza
names the idea; states what it is
Suppose that the conception of a triangle is given to a philosopher and that he is required to discover, by the philosophical method, what relation the sum of its angles bears to a right angle. He has nothing before him but the conception of a figure enclosed within three right lines, and, consequently, with the same number of angles.
Immanuel Kant, The Critique of Pure Reason
names the idea; states what it is; opens on it
(57) Wherefore we make one and the same affirmation concerning God when we say that He has from eternity decreed that three angles of a triangle are equal to two right angles, as when we say that He has understood it.
Benedictus de Spinoza, Theologico-Political Treatise — Part 1
names the idea; states what it is
Related ideas
- Princeton University Press
- Springer Science+Business Media
- If and only if
- Geometry
- Nature
- Euclid's Elements
- Tetrahedron
- Theorem
- Parallel postulate
- Oxford University Press
- Wiley (publisher)
- Prentice Hall
- Non-Euclidean geometry
- Cambridge University Press
- Euclidean geometry
- Isosceles triangle
- Ellipse
Place in the tradition
Part of Philosophy of science.