Final answer:
The United States gained territory in the early 19th century primarily through war, such as the Mexican-American War and the War of 1812.
Step-by-step explanation:
The major way the United States gained territory in the early 19th century was through war. One example of this is the Mexican-American War, which took place from 1846 to 1848 and resulted in the U.S. acquiring substantial territory in the Southwest, including California and Texas. Another example is the War of 1812, which led to the U.S. gaining control of present-day Florida.