Root system inner product is the Killing form. The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer. This is because formula_32 and formula_33 are both integers, by assumption, and formula_34 Since formula_35, the only possible values for formula_36 are formula_37 and formula_38, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of "α" other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2"α" or −2"α", are
Reuleaux triangle any curve of given constant width. This area is where "s" is the constant width. One method for deriving this area formula is to partition the Reuleaux triangle into an inner equilateral triangle and three curvilinear regions between this inner triangle and the arcs forming the Reuleaux triangle, and then add the areas of these four sets. At the other extreme, the curve of constant width that has the maximum possible area is a circular disk, which has area formula_2. The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°.