Triangle

A triangle is a polygon defined by its three sides and three vertices (the points at which sides meet). A triangle with vertices A, B, and C is typically denoted as triangle ABC. Any three points define both a unique triangle and a unique plane, in Euclidean geometry. (Non-planar triangles are those that exist along a non-planar surface, such as the spherical triangles defined by three points on the surface of a sphere, in spherical geometry.)

Along with the circle, the triangle is one of the most important shapes in geometry, trigonometry, and topology. Some of the earliest mathematical discoveries deal with triangles—the most familiar example is the Pythagorean theorem formulated by the ancient Greek, Pythagoras—and because of its natural resistance to lateral pressure, the shape is still critical in construction, as each side is supported by the other two. Many familiar types of triangles are defined according to the sizes of their internal angles or by the relative lengths of their sides. An angle is said to be opposite a side (or edge) C when it is formed by the other two sides, A and B.

Overview

The angles of a triangle always add up to exactly 180 degrees. Triangles described according to the relative lengths of their sides include equilateral triangles (discussed in more depth below), isosceles triangles (in which two sides are equal in length, and consequently the two angles opposite these sides are also equal), and scalene triangles (in which all sides are of different lengths, and thus no two angles are equal either). Triangles described according to their angles include right triangles (in which an interior angle is exactly 90 degrees—that is, a right angle) and oblique triangles (all non–right triangles). Oblique triangles are further categorized as obtuse triangles (or obtuse-angled triangles), in which one interior angle is greater than 90 degrees, and acute triangles, in which all angles are less than 90 degrees.

Two triangles are similar when each angle of one triangle has the same measure as the corresponding angle in the other triangle, or when corresponding sides in each triangle have lengths in the same ratio. (This can be used to find the values of a side or angle when two triangles are known to be similar.) They are said to be congruent when they have exactly the same size—each angle having the same measure as its corresponding angle—and each corresponding side is the exact same length.

A common math problem for students is the construction of a triangle. When the lengths of all three sides are known, students can use a compass and a straightedge to construct the triangle by creating a line segment and using the compass to create arcs with lengths equal to the other two sides, and finding where they intersect to determine the third vertex.

The area of a triangle is one-half the base times the height, or 1/2(bh), where the base can be considered to be any side, and height is the length of a line perpendicular from the vertex opposite that side to the line containing that side.

The exterior angles of a triangle are the three angles that are supplementary to each of its interior angles. Two angles are supplementary when their sum is a straight line—that is, when they add up to 180; thus, each exterior angle x is 180 – y, where y is an interior angle of the triangle. (Furthermore, the sum of any two interior angles in a triangle is always supplementary to the remaining interior angle.) Consequently, because the interior angles always add up to 180 degrees, the exterior angles of a triangle always add up to 360 degrees.

Special Right Triangles

A right triangle is one in which one angle is a right angle—that is, 90 degrees. The side opposite the right angle is the hypotenuse; the other sides are usually called the legs or, less commonly, the catheti (singular cathetus). Its area is 1/2(ab), where a and b are the triangle’s legs.

Special right triangles are those that possess some feature or features that simplify the formulae or calculations associated with triangles. These are usually divided into two groups: angle-based and side-based. Angle-based special right triangles include the 45-45-90 triangle, with interior angles of those values, which has the smallest ratio of c to a + b of any right triangle (√2/2). The 45-45-90 triangle is the only right triangle that is also an isosceles triangle, in Euclidean geometry.

The 30-60-90 right triangle similarly possesses interior angles of those values, and is noteworthy because its angles are in a 1:2:3 ratio, and its sides are in the ratio 1:√3:2, which simplifies many calculations.

Side-based special right triangles include Pythagorean triples, which are right triangles whose side lengths are all integers. Pythagorean triples are also noteworthy in that at least one of their angles, expressed in degrees, is an irrational number. Common Pythagorean triples include 3:4:5, 5:12:13, 8:15:17, and 7:24:25.

Equilateral Triangle

An equilateral triangle is one in which all three sides are equal and thus is equiangular as well, meaning that all three internal angles are congruent (that is, each is exactly 60 degrees). The equilateral triangle is a subtype of regular polygons: a polygon that, in Euclidean geometry, has both angles of equal measure and sides of equal length. It is the regular polygon with the smallest number of sides and angles, and is a convex regular polygon, though overlapping equilateral triangles may also be used to construct regular star polygons of n vertices, where n is a multiple of 3.

The area of an equilateral triangle is √3/4 (approximately 0.433) times the square of the length of one of its sides. The height, or altitude, from any side is (√3/2) times the length of one of its sides.

Pascal’s Triangle

Pascal’s triangle is not an actual triangle but a triangular arrangement of the binomial coefficients. Named for French mathematician Blaise Pascal, it has actually been independently discovered at various times and places in history, and had been known in India and China before Pascal’s discovery. The rows of Pascal’s triangle are arranged so that each has one more numeral than the row above it:

The triangle possesses various special properties. For instance, each number in the triangle is the sum of the two directly above it.

Theorems About Triangles

The best-known theorem about triangles is the Pythagorean theorem, which states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs, or a² + b² = c². (If a, b, and c are integers, the triangle is called a Pythagorean triangle and its side lengths a Pythagorean triple.) Named for Pythagoras, the 6^th century BCE Greek mathematician credited with its first proof, the theorem has been independently discovered by other cultures, including (at a minimum) the ancient cultures of India and China. It is believed that more proofs have been provided for the Pythagorean theorem than for any other proof in mathematics, including both geometry and algebraic proofs, as well as generalizations to higher-dimensional or non-Euclidean spaces.

Other common theorems include various congruence theorems, which state that if, for instance, the three sides of one triangle (or two sides and the included angle, or two angles and the included side) are congruent to those of another triangle, the triangles themselves are congruent. Similarity theorems state that if two angles of one triangle are congruent to two angles of another, the two triangles are similar. Other theorems deal, like the Pythagorean theorem, specifically with right triangles. The altitude rule says that the altitude to the hypotenuse of a right triangle is the mean proportional between the segments it divides the hypotenuse into.

Napoleon’s theorem is credited to Napoleon Bonaparte, but this may be apocryphal. It states that when equilateral triangles are constructed on the sides of any triangle, the centers of those equilateral triangles themselves define a fourth equilateral triangle.

Bibliography

Aigner, Martin, and Gunter M. Ziegler. Proofs from the Book. New York: Springer, 2014.

Hanna, Gila. Explanation and Proof in Mathematics. New York: Springer, 2014.

Millman, Richard, Peter Shiue, and Eric Brendan Kahn. Problems and Proofs in Numbers and Algebra. New York: Springer, 2015.

Perrin, Daniel. Algebraic Geometry: An Introduction. New York: Springer, 2008.