Types of triangles

Classifying by sides gives three types. An equilateral triangle has all three sides equal, which forces all three angles to equal 60°. An isosceles triangle has at least two equal sides, and the angles opposite those sides — the base angles — are equal to each other. A scalene triangle has three different side lengths and, consequently, three different angle measures. A useful companion rule is that the largest angle always sits opposite the longest side.

Classifying by angles gives three more types. An acute triangle has all three angles less than 90°. A right triangle has exactly one 90° angle, with the side opposite it called the hypotenuse. An obtuse triangle has one angle greater than 90°. A triangle can have at most one right or obtuse angle, because the three interior angles always sum to 180°.

The two classifications combine. A 45-45-90 triangle is isosceles and right; its sides are in the ratio 1 : 1 : √2. A 30-60-90 triangle is scalene and right, with sides in the ratio 1 : √3 : 2. An equilateral triangle is always acute. Right triangles satisfy the Pythagorean theorem, a² + b² = c², where c is the hypotenuse. And any three lengths can form a triangle only if they satisfy the triangle inequality: the sum of any two sides must exceed the third.

Triangle classification is heavily tested. The AMC 8 uses it in geometry problems that hinge on spotting an isosceles pair or an equilateral sub-triangle. The ACT and GRE both rely on the special right triangle ratios to let you find a missing side without a calculator, and both test the angle sum, the isosceles base-angle rule, and the area formula A = ½ · base · height.