Special right triangle are in the ratio 1 : 2 : 3 and respectively measure 30° (), 60° (), and 90° (). The sides are in the ratio 1 : : 2. The proof of this fact is clear using trigonometry. The geometric proof is: The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if "α", , are the angles in the progression then the sum of the angles = 180°. After dividing by 3, the angle must be 60°. The right angle
Skinny triangle based on the small-angle approximations; and, when formula_5 is in radians. The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1; The term formula_7 represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus "θ") are equal to "π". Applying the small angle approximations to the law of sines above results in; the desired result. This result is equivalent to assuming that the length of the base of