Tensor principal component analysis (TPCA), as a data dimensionality reduction algorithm aimed at representing high-dimensional tensor data in low dimensional subspaces, has been widely applied in multiple machine learning fields. However, the L₁-norm loses the rotation invariance and the existing TPCA algorithms adopt a single optimization objective, which only optimizes the projection distances and ignores the optimization of the error tensors. Thus, even though these algorithms have a certain degree of robustness, they still perform weakly. To address these issues, this paper proposes a ratio model for dual-objective optimization. This model is inspired by the formula for the area of a right-angled triangle, which optimizes the height on the hypotenuse to achieve dual-objective optimization with maximum projection distances and minimum reconstruction errors, called the area projection model. Then, based on the projection model, this article adopts a preprocessing technique of blocking recombination and proposes a block tensor PCA with F-norm based on area projection (area-BTPCA-F) algorithm. This algorithm not only preserves rotation invariance, but also fully considers error tensors. In response to noise information, blocking recombination technique has greatly improved the robustness of the algorithm. Finally, experiments on six color datasets with different noise validate the proposed algorithm, showing improvements in average reconstruction error (ARCE) and classification rate. The algorithm demonstrates strong robustness compared to other existing TPCA algorithms.