It is known that the Born-Infeld (BI) equation \[ (1 + u_ x^ 2) u_{tt} - 2u_ t u_ xu_{tx} - (1-u_ t^ 2) u_{xx} = 0, \] which is an old integrable model of the classical field theory, can be rewritten in the form of a system of two first-order nonlinear hyperbolic equations. This representation allows one to construct a natural Hamiltonian representation of the BI equation, as well as to solve it by means of the characteristics method. In this work, a transformation is found between the BI equation and the Monge-Ampère equation, \[ u_{tt} u_{xx}-u_{tx}^ 2 = - 1, \] which simultaneously produces representation of the latter equations in the form of a nonlinear hyperbolic system.