Fins are surface extensions used to increase the rate of heat transfer from a heated surface to the surrounding cooler fluid or from a heated fluid to the surface. In this article, we describe new families of straight and pin fins for which the temperature distribution along the fin as well as fin effectiveness and efficiency can be computed in closed form. The profile of the new straight fin is a circular arc centered on the fin’s longitudinal axis, while the profile of the new pin fin is a more complex concave parametric curve. Although the pattern of temperature distribution in these fins is remarkably similar to that in the classic straight and pin fins of constant thickness, the geometry of the new fins is strikingly different; in particular, the maximum length of the new fins is finite. We also discovered that both classic fins of constant thickness and novel fins described in this work have the following equivalence property: if the ratio of the heat transfer coefficient on the fin’s tip to that on the lateral surface is not too high, then the temperature distribution for a fin with a non-adiabatic boundary condition at the tip is identical to that for a longer fin of the same kind with adiabatic tip. Our analysis of heat transfer by fins is only based on standard homogeneity, steady-state, and one-dimensionality assumptions. In particular, we completely dispense with the “length-of-arc” assumption that underlies most of the previous works.