# Triangle Classification

The basic elements of any triangle are its *sides* and *angles*. Triangles are classified depending on relative sizes of their elements.

As regard their sides, triangles may be

(all sides are different)**Scalene**(two sides are equal)**Isosceles**(all three sides are equal)**Equilateral**

And as regard their angles, triangles may be

(all angles are acute)**Acute**(one angle is right)**Right**(one angle is obtuse)**Obtuse**(all angles are equal)**Equiangular**

A triangle is *scalene* if all of its three sides are different (in which case, the three angles are also different). If two of its sides are equal, a triangle is called *isosceles*. A triangle with all three equal sides is called *equilateral*. S. Schwartzman's *The Words of Mathematics* explain the etymology (the origins) of the words. The first two are of Greek (and related) origins; the word "equilateral" is of Latin origin:

**scalene** (adjective): from the Indo-European root *skel-* "to cut." Greek *skalenos* originally meant "stirred up, hoed up." When a piece of ground is stirred up, the surface becomes "uneven," which was a later meaning of *skalenos*. A scalene triangle is uneven in the sense that all three sides are of different lengths. The scalene muscles on each side of a person's neck are named for their triangular appearance. A scalene cone or cylinder is one whose axis is not perpendicular to its base; opposite elements make "uneven" angles with the base.

**isosceles** (adjective): from Greek *isos* "equal", of unknown prior origin, and *skelos* "leg". The Indo-European root *(s)kel-* "curved, bent" is found in *scoliosis* and *colon*, borrowed from Greek. In geometry, an isosceles triangle or trapezoid has two equal legs. It may seem strange that the root means "bent" even though the sides of a triangle or trapezoid are straight, but each leg is bent relative to the adjoining legs.

**equilateral** (adjective): from Latin *æquus* "even, level," and *latus*, stem *later-*, "side," both of uncertain origin. Related borrowings from Latin are *bilateral* and *multilateral*. In geometry, equilateral triangle is one in which all sides are equal in length.

This is how the two approaches are distinguished with Venn diagrams:

As regard the angles, a triangle is *equiangular* if all three of its angles are equal. Very early in the *Elements* (I.5 and I.6) Euclid showed that in an isosceles triangle the base angles are equal and, conversely, the sides opposite equal angles are equal. From here, for a triangle, the properties of being equilateral and equiangular are equivalent, and the latter is seldom mentioned. (For a polygon with the number of sides greater than 3 the equivalence no longer holds.)

In Euclidean geometry, the sum of the angles in a triangle equals 180°. It follows that a triangle may have at most one obtuse or even right angle. (This also follows from the Exterior Angle Theorem.) If one of the angles in a triangle is obtuse, the triangle is called *obtuse*. A triangle with one right angle is *right*. Otherwise, a triangle is *acute*; for all of its angles are acute. (All the definitions are naturally exclusive. There is no possible ambiguity.)

The following diagram summarizes all possible triangle configurations. The types of triangles:

I came across this diagram in [Jacobs, p. 149] who credits *First Steps in Geometry* by G. A. Wentworth and G. A. Hill (Ginn, 1901).

## References

- H. R. Jacobs,
*Geometry*, 3^{rd}edition, W. H. Freeman and Company, 2003 - S. Schwartzman,
*The Words of Mathematics*, MAA, 1994

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