The aim of this paper is to explore the optimal geometric form of a building that minimizes direct solar irradiation incident on the envelope, and to find useful guidelines for building designers at the early decision-making stage. To achieve this goal, the mathematical theory of Calculus of Variation is used to find the theoretical optimal solutions and, based on this analytical approach, a simplified model to calculate annual solar heat gains is introduced. The benefit of using a totally analytical optimization is to provide simple and universal geometrical rules applicable in a general scenario, despite of using a numerical optimization that is more suitable to solve specific and complex cases by considering all of the aspects involved, e.g. the internal structure of the building or the optical/thermal properties of material used. The optimization is applied to various cases with different constraints. Our results suggest some rules to follow, even in the general case whereby the plan is fixed by the building's designer, to achieve a reduction in the total amount of direct solar irradiation without necessarily reducing the annual solar heat gains. For this reason our results could be applied not only in hot regions but also in mild and cold ones. (C) 2013 Elsevier Ltd. All rights reserved.