The purpose of this paper is to construct a Quasi-volume-filling surface and study its properties. We start with the construction of a volume-filling surface, the Pólya surface, based on Pólya's curve, by rotating the Pólya's curve in 3-dimensional space. Then we construct a Quasi-space-filling curve in 2-dimensions, the Quasi- Pólya curve, which approximates the Pólya's curve and fills a triangle up to a residual small surface of arbitrary size. We prove that the Quasi-Pólya curve satisfies the open set condition, and there exists a unique invariant (self-similar) measure consistent with the normalized Hausdorff measure on it. Moreover, the energy form constructed on Quasi-Pólya curve is proved to be a closed & regular form, and we prove that the Quasi-Pólya curve is a variational fractal in the end. Next, we use the same idea, by rotating the Quasi-Pólya curve in 3-dimensional space, to construct the Quasi-Pólya surface, which is a Quasi-volume-filling surface and approximates to Pólya surface in some sense.