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A triangle consists of three line segments forming a closed figure. In general, closed figures composed of line segments are called "polygons," a word whose etymology contains the Greek words for "many" and "sides". Triangles with three equal sides (and hence three equal angles) are called "equilateral". If only two sides (and the two angles opposite those sides) are equal, the triangle is "isosceles". A scalene triangle has three unequal sides.

Right triangles have one ninety-degree angle. Because ninety degree angles are important in architecture and construction, they have a special name- "right angles". Right angles have an interesting relationship between their sides known as the Pythagorean theorem. Square the lengths of each side of a right triangle, and you'll find that the squares of the shorter two sides add up to the square of the longest side (called the "hypotenuse".)

The Pythagorean theorem is used in construction and home improvement in order to check whether or not something is "square". If, for example, two boards forming an L-shaped joint are "square", then the squares of the lengths of the two boards should equal the square of the diagonal line connecting the ends of the boards. If the numbers don't come out right, then the join is not square.

The branch of mathematics known as trigonometry deals with right triangles and the relationships between their angles and sides. An understanding of trigonometry is best achieved by studying right triangles in depth before moving on to other applications. Besides the basic surveying and building applications, trigonometry touches every area of the physical sciences. The sine function, which can be thought of as the ratio of the side opposite an angle to the hypotenuse of a right triangle containing that angle, is used to model electromagnetic radiation, harmonic motion, and circular motion. In classical Newtonian mechanics, the sine and cosine functions allow forces to be analyzed in component form, greatly simplifying calculations.

The triangle is a simple geometric figure, but illustrates mathematical principles of immense power. Many students question the value of mathematics, mistakenly equating abstraction with frivolity. In reality, it is that very abstraction that makes mathematics so versatile. Something as simple as a triangle or as complicated as an overlapping pattern of sound waves can be analyzed with the same mathematical principles.

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