There is growing interest in platform trials that allow for adding of new treatment arms as the trial progresses as well as being able to stop treatments part way through the trial for either lack of benefit/futility or for superiority. In some situations, platform trials need to guarantee that error rates are controlled. This paper presents a multi-stage design, that allows additional arms to be added in a platform trial in a preplanned fashion, while still controlling the family-wise error rate, under the assumption of known number and timing of treatments to be added, and no time trends. A method is given to compute the sample size required to achieve a desired level of power and we show how the distribution of the sample size and the expected sample size can be found. We focus on power under the least favorable configuration which is the power of finding the treatment with a clinically relevant effect out of a set of treatments while the rest have an uninteresting treatment effect. A motivating trial is presented which focuses on two settings, with the first being a set number of stages per active treatment arm and the second being a set total number of stages, with treatments that are added later getting fewer stages. Compared to Bonferroni, the savings in the total maximum sample size are modest in a trial with three arms, <1% of the total sample size. However, the savings are more substantial in trials with more arms.