The way the text explained it we can actually calculate SSA, SSW, and SST all separately; however, as you have explained, SSA + SSW = SST. The text showed a couple of examples where they actually calculated SST and SSW first. They then subtract SST-SSW which equal SSA. If we look at the calculations, assuming we are calculating this by hand, it is actually much easier to SSA since this formula is fairly simple. That leaves us the need to only calculate SSW. In reality, we do not need to calculate SST to find the F value, all we need is SSA and SSW. We use SSA and SSW to get our MSA and MSW values. We use the MSA and divide it by MSW to get the F statistic.