Recently, mathematics has a wide influence on architectural design and performance. It can effectively help in solving design problems and promoting geometrical shapes efficiency. Consequently, the environmental performance has also widely developed with the involving of advanced mathematical branches and computational software. The main aim of the paper is to study the importance of using mathematics in raising the building's environmental performance. Therefore, the research methodology starts by clarifying the mathematics' branches and their application in architecture, then carries out an analysis for different case studies that used mathematics effectively in developing environmental performance. And lastly, apply one of the mathematical optimization techniques on a virtual prototype to illustrate how mathematics can effectively promote the building's environmental efficiency.

Sources of energy are the most important challenges facing the world in this century. The global interest in energy and its relation to the construction sector that consider the most energy consuming sectors, which led to a search for methods of energy conservation. Research assumes a methodology to apply the modern technology in buildings and gaining advantages of developed building materials, outer envelopes, and conserve energy systems to reduce energy waste and increase its efficiency by integrating natural and renewable energy applications. This methodology could be applied to buildings during the design phase to achieve energy rationalization.