Abstract The analysis and optimization of flat plate fins of constant thickness and straight base has been conducted for different fin’s shapes. Performance of the fins is quantified through effectiveness and expressed as a function of fin’s shape, width, and area. A linear piecewise function with varying number of evenly spaced sections is used to generate shapes with different number of degrees of freedom, which are classified as constrained- or unconstrained-base depending on the width of the fin at the constant temperature base location. For one- and two-degrees of freedom shapes, a variety of rectangular, triangular, trapezoidal, and rhomboidal geometries are considered and optimized. For more than two-degrees of freedom, more complex resulting shapes are also considered. By adjusting the value of the corresponding shape parameters, the best possible distributions of available area to maximize heat transfer are obtained, which produce significant improvements favored by smaller width and larger area. Unconstrained-base performs better than constrained-base configurations as they allow the distribution of a larger area close to the high-temperature base. Optimal shape of unconstrained-base fins is in general convergent with non-zero wide tip, while, for constrained-base fins, it is divergent in the first linear section from the base and convergent in the remaining ones. For increasing number of degrees of freedom the optimized shapes tend to resemble natural structures while effectiveness increases asymptotically to a limit for constant width and area. Besides determining the optimal configurations, consideration has been made about the optimal regions where a variety of shapes produce virtually the same effectiveness of the absolute maximum, which can be used to design fins with almost maximum performance but having simpler shapes or being functional under possible space restrictions. The dimensionless model and the systematic analysis proposed in this work are not only appropriate to study a wide range of flat plate fins but also can be implemented to analyze and perform optimization over other type of fins and fins configurations.