Correct reconstruction of macromolecular structure by cryo-electron microscopy (cryo-EM) relies on accurate determination of the orientation of single-particle images. For small (<100 kDa) DNA-binding proteins, obtaining particle images with sufficiently asymmetric features to correctly guide alignment is challenging. We apply DNA origami to construct molecular goniometers-instruments that precisely orient objects-and use them to dock a DNA-binding protein on a double-helix stage that has user-programmable tilt and rotation angles. We construct goniometers with 14 different stage configurations to orient and visualize the protein just above the cryo-EM grid surface. Each goniometer has a distinct barcode pattern that we use during particle classification to assign angle priors to the bound protein. We use goniometers to obtain a 6.5-Å structure of BurrH, an 82-kDa DNA-binding protein whose helical pseudosymmetry prevents accurate image orientation using traditional cryo-EM. Our approach should be adaptable to other DNA-binding proteins as well as small proteins fused to DNA-binding domains.