Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. 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 edges; 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.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, a triangle with two sides having the same length is an isosceles triangle, and a triangle with three different-length sides is a scalene triangle. A triangle in which one of the angles is a right angle is a right triangle, a triangle in which all of its angles are less than that angle is an acute triangle, and a triangle in which one of it angles is greater than that angle is an obtuse triangle. These definitions date back at least to Euclid. In man-made construction, the isosceles triangles may be found in the shape of gables and pediments, and the equilateral triangle can be found in the yield sign. The faces of the Great Pyramid of Giza are sometimes considered to be equilateral, but more accurate measurements show they are isosceles instead. Other appearances are in heraldic symbols as in the flag of Saint Lucia and flag of the Philippines.
Each triangle has many special points inside it, on its edges, or otherwise associated with it. They are constructed by finding three lines associated symmetrically 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 points 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. A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, 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. Thales' theorem implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. 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.
In the Euclidean plane, area is defined by comparison with a square of side length 1, which has area 1. There are several ways to calculate the area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side (the base) times the corresponding altitude. This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base b and height h. Heron's formula, named after Heron of Alexandria, is a formula for finding the area of a triangle from the lengths of its sides a, b, c. Letting s be the semiperimeter, the area equals the square root of s(s-a)(s-b)(s-c). Unlike a rectangle, which may collapse into a parallelogram from pressure to one of its points, triangles are sturdy because specifying the lengths of all three sides determines the angles. Therefore, a triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. For this reason, structural quadrilaterals are often built with a diagonal joint to split it into two rigid triangles.
Triangulation means the partition of any planar object into a collection of triangles. For example, in polygon triangulation, a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon. In the case of a simple polygon with n sides, there are n-2 triangles that are separated by n-3 diagonals. Triangulation of a simple polygon has a relationship to the ear, a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears. Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which gives a congruent triangle, or even by rescaling it to a similar triangle: Trilinear coordinates specify the relative distances of a point from the sides, so that coordinates alpha:beta:gamma indicate that the ratio of the distance of the point from the first side to its distance from the second side is alpha:beta, etc. Barycentric coordinates of the form u:v:w specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point.
A non-planar triangle is a triangle not embedded in a Euclidean space, roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in hyperbolic space and spherical geometry. A triangle in hyperbolic space is called a hyperbolic triangle, and it can be obtained by drawing on a negatively curved surface, such as a saddle surface. Likewise, a triangle in spherical geometry is called a spherical triangle, and it can be obtained by drawing on a positively curved surface such as a sphere. The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180 degrees. However, the sum of the internal angles of a hyperbolic triangle is less than 180 degrees, and for any spherical triangle, the sum is more than 180 degrees. In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90 degrees, adding up to a total of 270 degrees. By Girard's theorem, the sum of the angles of a triangle on a sphere is E/R^2, where E is the fraction of the sphere's area enclosed by the triangle.
Common questions
What is a triangle and how many sides does it have?
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners are called vertices while the sides connecting them are called edges.
How do you calculate the area of a triangle using base and height?
The area of a triangle equals one-half the product of height and base length. This formula can be proven by cutting up the triangle and an identical copy into pieces to form a rectangle.
What are the different types of triangles based on side lengths?
An equilateral triangle has all sides of the same length, an isosceles triangle has two sides of the same length, and a scalene triangle has three different-length sides. These definitions date back at least to Euclid.
Where can you find triangles in man-made construction and flags?
Isosceles triangles appear in gables and pediments while equilateral triangles appear in yield signs. Triangles also appear in heraldic symbols such as the flag of Saint Lucia and the flag of the Philippines.
How do angles differ between Euclidean hyperbolic and spherical triangles?
Internal angles of a Euclidean triangle always add up to 180 degrees. A hyperbolic triangle has internal angles summing to less than 180 degrees while a spherical triangle sums to more than 180 degrees.