Exercise-induced maximum aerobic metabolic rate (MMR) is related to the fractal dimension (D) of the self-similar vascular blood transport system by a whole body mass (Mw) power with exponent b of the form b = D/3. The principle of self-similarity of the vascular system is in agreement with each organ in the body having a major artery supplying it with blood from the heart and a major vein returning it. This implies that the whole body fractal vascular dimension D is also applicable to all organs or collections of organs such as the viscera and skeletal muscle. The principal reason that basal metabolic rate (BMR) and MMR scale with different power exponents to whole body mass is that MMR is due mainly to respiration in skeletal muscle during exercise and BMR to respiration in the viscera during rest. It follows, therefore, from the self-similarity of the vascular system that BMR is related to viscera mass (Mv) in the same way that MMR is related to muscle mass. Hence, BMR scales to Mv with exponent b and, additionally, if Mv scales to Mw with exponent d, then BMR will scale with Mw with exponent bd, where bd = b2 for b = d. Here this approach is justified by an assessment of the scaling of viscera and its components with total body mass. The applicability of fractal vascular scaling to the sum of visceral organ metabolic rate contributions is confirmed from organ tissue slices, mitochondrial surface areas, and blood oxygen transport. Estimates obtained from oxygen half-saturation partial pressure scaling exponents show that BMR scaling with b2 is of general occurrence between species.