K3BiO4 crystallizes in the triclinic P-1 space group. There are three inequivalent K sites. In the first K site, K(1) is bonded to two equivalent O(1), two equivalent O(2), and two equivalent O(3) atoms to form distorted KO6 pentagonal pyramids that share corners with two equivalent K(2)O5 trigonal bipyramids, corners with two equivalent Bi(1)O5 trigonal bipyramids, edges with three equivalent K(1)O6 pentagonal pyramids, edges with two equivalent Bi(1)O5 trigonal bipyramids, and edges with three equivalent K(2)O5 trigonal bipyramids. In the second K site, K(2) is bonded to one O(1), one O(3), one O(4), and two equivalent O(2) atoms to form KO5 trigonal bipyramids that share corners with two equivalent K(1)O6 pentagonal pyramids, corners with four equivalent Bi(1)O5 trigonal bipyramids, edges with three equivalent K(1)O6 pentagonal pyramids,  an edgeedge with one K(2)O5 trigonal bipyramid, and  an edgeedge with one Bi(1)O5 trigonal bipyramid. In the third K site, K(3) is bonded in a 6-coordinate geometry to one O(1), one O(2), two equivalent O(3), and two equivalent O(4) atoms. Bi(1) is bonded to one O(1), one O(2), one O(3), and two equivalent O(4) atoms to form BiO5 trigonal bipyramids that share corners with two equivalent K(1)O6 pentagonal pyramids, corners with four equivalent K(2)O5 trigonal bipyramids, edges with two equivalent K(1)O6 pentagonal pyramids,  an edgeedge with one K(2)O5 trigonal bipyramid, and  an edgeedge with one Bi(1)O5 trigonal bipyramid. There are four inequivalent O sites. In the first O site, O(1) is bonded in a 5-coordinate geometry to one K(2), one K(3), two equivalent K(1), and one Bi(1) atom. In the second O site, O(2) is bonded to one K(3), two equivalent K(1), two equivalent K(2), and one Bi(1) atom to form distorted OK5Bi octahedra that share corners with three equivalent O(3)K5Bi octahedra, corners with two equivalent O(4)K3Bi2 trigonal bipyramids, edges with two equivalent O(2)K5Bi octahedra, edges with three equivalent O(3)K5Bi octahedra, and edges with two equivalent O(4)K3Bi2 trigonal bipyramids. The corner-sharing octahedral tilt angles range from 13-23°. In the third O site, O(3) is bonded to one K(2), two equivalent K(1), two equivalent K(3), and one Bi(1) atom to form distorted OK5Bi octahedra that share corners with three equivalent O(2)K5Bi octahedra, corners with three equivalent O(4)K3Bi2 trigonal bipyramids, edges with two equivalent O(3)K5Bi octahedra, edges with three equivalent O(2)K5Bi octahedra, and edges with two equivalent O(4)K3Bi2 trigonal bipyramids. The corner-sharing octahedral tilt angles range from 13-23°. In the fourth O site, O(4) is bonded to one K(2), two equivalent K(3), and two equivalent Bi(1) atoms to form distorted OK3Bi2 trigonal bipyramids that share corners with two equivalent O(2)K5Bi octahedra, corners with three equivalent O(3)K5Bi octahedra, edges with two equivalent O(2)K5Bi octahedra, edges with two equivalent O(3)K5Bi octahedra, and edges with two equivalent O(4)K3Bi2 trigonal bipyramids. The corner-sharing octahedral tilt angles range from 13-80°.
K3BiO4 crystallizes in the triclinic P-1 space group. There are three inequivalent K sites. In the first K site, K(1) is bonded to two equivalent O(1), two equivalent O(2), and two equivalent O(3) atoms to form distorted KO6 pentagonal pyramids that share corners with two equivalent K(2)O5 trigonal bipyramids, corners with two equivalent Bi(1)O5 trigonal bipyramids, edges with three equivalent K(1)O6 pentagonal pyramids, edges with two equivalent Bi(1)O5 trigonal bipyramids, and edges with three equivalent K(2)O5 trigonal bipyramids. There is one shorter (2.77 Å) and one longer (2.83 Å) K(1)-O(1) bond length. There is one shorter (2.72 Å) and one longer (2.78 Å) K(1)-O(2) bond length. There is one shorter (2.82 Å) and one longer (2.94 Å) K(1)-O(3) bond length. In the second K site, K(2) is bonded to one O(1), one O(3), one O(4), and two equivalent O(2) atoms to form KO5 trigonal bipyramids that share corners with two equivalent K(1)O6 pentagonal pyramids, corners with four equivalent Bi(1)O5 trigonal bipyramids, edges with three equivalent K(1)O6 pentagonal pyramids,  an edgeedge with one K(2)O5 trigonal bipyramid, and  an edgeedge with one Bi(1)O5 trigonal bipyramid. The K(2)-O(1) bond length is 2.60 Å. The K(2)-O(3) bond length is 2.64 Å. The K(2)-O(4) bond length is 2.67 Å. There is one shorter (2.73 Å) and one longer (2.77 Å) K(2)-O(2) bond length. In the third K site, K(3) is bonded in a 6-coordinate geometry to one O(1), one O(2), two equivalent O(3), and two equivalent O(4) atoms. The K(3)-O(1) bond length is 2.80 Å. The K(3)-O(2) bond length is 3.16 Å. There is one shorter (2.75 Å) and one longer (2.77 Å) K(3)-O(3) bond length. There is one shorter (2.88 Å) and one longer (2.90 Å) K(3)-O(4) bond length. Bi(1) is bonded to one O(1), one O(2), one O(3), and two equivalent O(4) atoms to form BiO5 trigonal bipyramids that share corners with two equivalent K(1)O6 pentagonal pyramids, corners with four equivalent K(2)O5 trigonal bipyramids, edges with two equivalent K(1)O6 pentagonal pyramids,  an edgeedge with one K(2)O5 trigonal bipyramid, and  an edgeedge with one Bi(1)O5 trigonal bipyramid. The Bi(1)-O(1) bond length is 2.03 Å. The Bi(1)-O(2) bond length is 2.04 Å. The Bi(1)-O(3) bond length is 2.06 Å. There is one shorter (2.11 Å) and one longer (2.28 Å) Bi(1)-O(4) bond length. There are four inequivalent O sites. In the first O site, O(1) is bonded in a 5-coordinate geometry to one K(2), one K(3), two equivalent K(1), and one Bi(1) atom. In the second O site, O(2) is bonded to one K(3), two equivalent K(1), two equivalent K(2), and one Bi(1) atom to form distorted OK5Bi octahedra that share corners with three equivalent O(3)K5Bi octahedra, corners with two equivalent O(4)K3Bi2 trigonal bipyramids, edges with two equivalent O(2)K5Bi octahedra, edges with three equivalent O(3)K5Bi octahedra, and edges with two equivalent O(4)K3Bi2 trigonal bipyramids. The corner-sharing octahedral tilt angles range from 13-23°. In the third O site, O(3) is bonded to one K(2), two equivalent K(1), two equivalent K(3), and one Bi(1) atom to form distorted OK5Bi octahedra that share corners with three equivalent O(2)K5Bi octahedra, corners with three equivalent O(4)K3Bi2 trigonal bipyramids, edges with two equivalent O(3)K5Bi octahedra, edges with three equivalent O(2)K5Bi octahedra, and edges with two equivalent O(4)K3Bi2 trigonal bipyramids. The corner-sharing octahedral tilt angles range from 13-23°. In the fourth O site, O(4) is bonded to one K(2), two equivalent K(3), and two equivalent Bi(1) atoms to form distorted OK3Bi2 trigonal bipyramids that share corners with two equivalent O(2)K5Bi octahedra, corners with three equivalent O(3)K5Bi octahedra, edges with two equivalent O(2)K5Bi octahedra, edges with two equivalent O(3)K5Bi octahedra, and edges with two equivalent O(4)K3Bi2 trigonal bipyramids. The corner-sharing octahedral tilt angles range from 13-80°.