In order to explore the role of initial geometry in wound closure, we developed a new two-dimensional wound healing assay which we refer to as a sticker assay. Stickers are produced and attached to a tissue culture plate before cells are seeded into the centre of a dish. The stickers are then removed to leave a wound in the dish, which is an area with no cells. We performed a number of sticker assays using NIH 3T3 fibroblast cells and various initial wound shapes, including circles, squares, triangles and rectangles. By applying an edge detection method in ImageJ, we tracked the wound interface (and area) at discrete times up to 96h and found that different shaped wound closed at different rates. To explore these experimental results further, we applied two mathematical models, the first being a lattice-based random walk algorithm in two dimensions with nearest-neighbour exclusion, the second being a continuum-limit description based on the Fisher-Kolmogorov equation. By calibrating our models to the experimental data, we find that the parameter estimates for each initial wound shape are similar, suggesting that, while the closure rate is different for each initial wound shape, the underlying mechanisms that drive wound closure are the same. Finally, we revisit our continuum model by allowing for nonlinear degenerate diffusion of the type that occurs in the porous medium equation (charaterised by an index $n$). We explore the hypothesis that circular wound closure is self-similar and circular wound boundaries are unstable for an increasing number of $k$-fold symmetric perturbations as the index $n$ decreases. In particular, we highlight the consequences for wound closure with an initially rectangular wound, where both numerical and experimental results suggest that the wound becomes very long and thin in the limit that it closes.