How much land was added to the US under President James K. Polk?

Under James K. Polk, who ran for office on a platform of national expansion, the United States annexed Texas, settled the border between the Oregon Territory and British Canada, and received the Mexican Cession from Mexico. 

The annexation of Texas was a foregone conclusion by the time Polk entered office--indeed, Congress formalized the measure in the final days of the Tyler administration. The annexation would prove to be highly consequential, as disputes over the border between Mexico and Texas (as well as Polk's desire for Mexican lands in the North American Southwest) led to the Mexican-American War. American victory in this war led to the Mexican Cession, which included modern-day California, New Mexico, Arizona, Utah, and Nevada. 

Polk's settlement with Mexico, while falling far short of his demands for the border to be drawn at the 54'40 line, significantly expanded American territory. The lands formally recognized as belonging to the United States included modern-day Washington, Oregon, and Idaho.

In short, under James K. Polk, the United States witnessed unprecedented expansion, almost (with the small exception of the Gadsden Purchase from Mexico in the Southwest) filling out its current borders.