Special Right Triangles: 30-60-90 and 45-45-90

These two triangles show up constantly because they come from cutting simple shapes in half. Slice a square along its diagonal and you get a 45-45-90 triangle: the two legs are the square's equal sides, and the diagonal is the hypotenuse. Slice an equilateral triangle straight down the middle, from one vertex to the midpoint of the opposite side, and you get a 30-60-90 triangle: the original side becomes the hypotenuse, half of it becomes the shorter leg, and the altitude becomes the longer leg. Because the starting shapes are fixed (a square has equal sides and right angles, an equilateral triangle has three 60 degree angles), the resulting triangles always have the same side ratios, no matter their size.

For 45-45-90, the ratio is x : x : x root 2, where x is a leg and x root 2 is the hypotenuse. For 30-60-90, the ratio is x : x root 3 : 2x, where x is the side opposite the 30 degree angle (the shortest side), x root 3 is opposite the 60 degree angle, and 2x is the hypotenuse, opposite the right angle. These ratios let you find any side from any other side using multiplication or division, with no trig and no calculator needed.

This topic connects directly to right-triangle trig and to area and volume problems, since these ratios often hand you the exact height or diagonal a formula needs. Recognizing a 30-60-90 or 45-45-90 triangle buried inside a square, equilateral triangle, hexagon, or cone saves a full trig calculation.