Applications of the theorems; dot product

Let a be the longest side. If b² + c² > a², the cosine is positive, so the triangle is acute; if b² + c² = a², it is right-angled; if b² + c² < a², it is obtuse. A right or obtuse angle always lies opposite the longest side. From S = ½bc·sinA and the law of sines, sinA = a/(2R), we get S = abc/(4R), that is R = abc/(4S). When three sides are given the area comes from Heron's formula: with semi-perimeter p = (a + b + c)/2, S = √(p(p − a)(p − b)(p − c)). So from three sides we can find the area, and then the circumradius, without finding any angle.