A triangle with three equal sides is called an equilateral triangle. When only two of the sides are equal, it is called an isosceles triangle. If none of the sides is the same, the figure is called a scalene triangle. Equilateral triangles have equal angles, each 60 degrees. Isosceles triangles contain two equal angles, and a scalene triangle has no equal angles. Triangles can also be described in reference to 90-degree angles. An acute triangle is one with all the angles measuring less than 90 degrees; a right-angle triangle has one 90-degree angle, while an obtuse triangle has an angle greater than 90 degrees.
The ratios of the sides of a right triangle present an interesting case. The side directly opposite the right angle is called the hypotenuse. A famous ancient Greek mathematician called Pythagoras studied the relationship of the sides of a right angle triangle around 500 B.C. He found that the square of the hypotenuse is always equal to the sum of the squares of the lengths of the other two sides. This relationship is called the Pythagorean theorem.
The relationship of the sides of a right triangle is another noteworthy feature in trigonometry. In a right triangle, the hypotenuse is like a ladder leaning against a wall. The wall is the opposite side and the floor the adjacent side. The ratio of the opposite side to an angle over the hypotenuse is called the sine of the angle. The ratio of the adjacent of the angle over the hypotenuse is called the cosine of the angle. The tangent is the ratio of the opposite side to the adjacent side of the angle. These three ratios -- sine, cosine and tangent -- have useful applications in science and technology and even in nature.
Triangular shapes abound in nature -- for example, the edge of some crystals, flower petals, leaves and even bones. The Pythagorean theorem finds application in measurements of distances. The concepts of sine, cosine and tangent find applications in physics and engineering and many scientific equipment used in surveying and geographical positioning and navigation. The study the interaction of a right triangle with a circle is called trigonometry. For example, a triangle moving within a circle traces out a very symmetrical wave called a sine wave. This wave is useful in the study of motion in physics.