The daughter of a prosperous Parisian merchant, Germain showed an early interest in mathematics and from the age of thirteen read whatever texts she could obtain. Although the main higher education institutions were closed to her, she managed to acquire the lecture notes of the mathematician J. L. Lagrange, which he had delivered at the newly founded Ecole Polytechnique.

She also began to correspond with prominent mathematicians using the pseudonym Le Blanc and allowing them to assume that she was a man. She had been working on number theory and had begun to tackle the celebrated last theorem of Fermat: that there are no integers x,y,z,n, where n ≥ 2 such that

xⁿ + yⁿ = zⁿ

Germain made a major contribution to showing that the equation does not hold for the case in which n is equal to 5. She informed Gauss of the result but, typically, he failed to reply.

In 1809 Germain began to work on the theory behind the appearance of curious patterns formed by sand placed on vibrating plates. The phenomenon had first been described by E. F. Chladni who had demonstrated them to Napoleon in 1808. The emperor had been so intrigued that he had offered a one-kilogram gold medal to the first person to explain what are now known as Chladni's figures.

Germain submitted a solution in 1811 based on Euler's theory of elasticity. She was the only entrant but her work contained a number of errors. It did, however, provoke Lagrange to produce a corrected equation to derive the patterns theoretically. The competition was extended, and after two further attempts, Germain was finally awarded the prize in 1815. She published her work privately in 1821 as Recherches sur la théorie des surfaces élastiques (Researches on the Theory of Elastic Surfaces).

Sophie Germain developed breast cancer in 1829 and died two years later.