The planar scattering amplitudes of 4d N = 4 super Yang-Mills theory and 3d N=6 Chern-Simons theory exhibit a remarkable property known as dual conformal symmetry. Motivated by this, we investigate the consequences of dual conformal symmetry in six dimensions. We find that 6d dual conformal symmetry fixes the structure of the one-loop 4-point amplitude and suggests a Lagrangian with more than two derivatives. Moreover, we obtain a similar structure by generalizing the Alday-Maldacena solution for a string in Anti-de Sitter space to a 2-brane ending on a pillow-shaped surface in the boundary whose seams correspond to a null-polygon, which suggests an all-loop formula for the 4-point amplitude similar to that of N=4 super Yang-Mills theory.