In this thesis, we will prove a very important theorem in real analysis called The Area formula for the Hausdorff Measure. This theorem is an extension of the well known theorem the Change of variables formula for the Lebesgue measure. In this thesis, we will define the Hausdorff measure and prove some of its properties. We will also define Lipschitz functions and prove some of its properties also. Then, we continue the thesis by proving all the lemmas needed to finalize the proof of the Area formula for the Hausdorff measure. Finally, we finish this thesis by showing three applications of the ...