Triangle

A triangle (also written as three-angle or three-sided) is a geometric shape made up of three straight line segments that are not all collinear, and three line segments (sides) that connect those points. The place where two sides meet is called a vertex or angle. The triangle is the polygon with the fewest sides, and it is the foundation for other areas of geometry, since any larger polygon can be divided into triangles.

The most well-known property of triangles in Euclidean geometry is that the sum of the three interior angles equals 180 degrees. This property depends on the parallel postulate; non-Euclidean geometries—such as those on the surface of a sphere—may have angle sums greater than or less than 180 degrees.

Classification

Triangles are classified in two ways. By sides: equilateral, in which all three sides are equal; isosceles, in which two sides are equal; and scalene, in which no sides are equal. By angles: acute, in which all three angles are less than 90 degrees; right, in which one angle is 90 degrees; and obtuse, in which one angle is greater than 90 degrees.

Area and measurement

The area of a triangle is calculated as half the base multiplied by the height: A = (1/2) × b × h, where b is a chosen side and h is the perpendicular distance from that side to the opposite vertex. Other methods exist, such as Heron's formula, which uses only the three sides, and the formula A = (1/2)ab·sin(C), which uses an angle and two sides. The perimeter of a triangle is the sum of the three sides.

Pythagoras' theorem and trigonometry

A right triangle is calculated using Pythagoras' theorem: the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. The right triangle also forms the basis of trigonometry, where sine, cosine, and tangent are defined as ratios of the sides. General triangles are handled using the law of sines and the law of cosines.

Applications

The triangle has wide application in construction and engineering, because the triangular frame does not change shape when its sides are fixed—a property called rigidity. This is why triangles are used in trusses and roofs. Similarly, in computer graphics, three-dimensional images are decomposed into networks of triangles.