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).

triangle, in music, percussion instrument consisting of a steel rod bent into a triangle, open at one angle, and struck with a steel rod. Only since the end of the 18th cent. has it been an orchestral instrument, although it appeared in Europe much earlier. Its tinkling sound is of indefinite pitch and therefore blends with whatever harmonies the orchestra produces.

Triangle Waist Company, often called the Triangle Shirtwaist Co., manufacturers of women's cotton and linen blouses. Located in lower Manhattan in the early 20th cent., on Mar. 25, 1911 it was the site of New York City's worst factory fire. The company, which occupied the top three floors of the 10-story Asch Building, employed some 500 young seamstresses, mainly Jewish and Italian immigrants, and less than 100 men. The fire began on the eighth floor at about 4:45 P.M.; fed by burning cloth, it became a conflagration. Although hindered by inward-opening doors that slammed shut in the crush, most of those on the eighth and tenth floors managed to escape, but on the ninth the rear door, bolted to prevent theft, could not be opened, and after the fire escape collapsed most were trapped. Clothes and hair ablaze, many women jumped to their deaths. Fire companies could do little, as neither water from their hoses nor their ladders reached above the seventh floor and their safety nets ripped with the weight of so many. In less than 15 minutes 146 died, nearly all women.

The company's owners were tried for manslaughter, but acquitted (1914), and their liability was limited to $75 in damages paid to 23 of the victims' families, awarded after a civil suit. The outcry occasioned by the fire, however, led to important reforms. The Factory Investigating Commission (headed by Robert F. Wagner and Alfred E. Smith), the Bureau of Fire Investigation, and the Fire Department's Fire Prevention Division were all established later in 1911. The ultimate result of their investigations were new labor, health, and fire safety laws, which, among other things, mandated outward-opening doors, sprinkler systems, fire drills, and regular building inspections, and forbade locked doors during working hours. The fire also led to increasingly successful labor-union organizing in city factories and sweatshops, particularly by the International Ladies Garment Workers Union and, more broadly, to a liberal and reformist movement within the Democratic party.

Research Triangle Park, research, business, medical, and educational complex situated in central North Carolina. It has an area of 6,900 acres (2,795 hectares) and is 8 × 2 mi (13 × 3 km) in size. Named for the triangle formed by Duke Univ. in Durham, the Univ. of North Carolina in Chapel Hill, and North Carolina State Univ. in Raleigh, it is one of the largest research complexes in the United States. The park was created in 1959 by leaders from business and academia; by the late 1990s it was home to nearly 140 organizations, more than a hundred of which were related to research and development, and employed about 50,000 people. Companies represented there engage in high-technology research, development, and manufacturing in such areas as the health sciences, pharmaceuticals, computers, optics, and many other for-profit and nonprofit enterprises. Development of the park, which has no residential facilities, has caused surrounding communities to mushroom and has created some local transportation problems.

Bermuda Triangle, area in the Atlantic Ocean off Florida where a number of ships and aircraft have vanished. Also known as the Devil's Triangle, it is bounded at its points by Melbourne, Fla.; Bermuda; and Puerto Rico. Storms are common in the region, and investigations to date have not produced scientific evidence of any unusual phenomena involved in the disappearances.

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.

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Triangular section of the North Atlantic Ocean whose boundaries are usually said to be Bermuda, the southern U.S. coast, and the Greater Antilles. The region attracted international attention after numerous planes and ships were said to have mysteriously disappeared there. Reports of unnatural occurrences were popularized, but by the late 20th century much of the myth surrounding the Bermuda Triangle had been dispelled.

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