In recent years, a technique has been developed to compute the conformal radii of random domains defined by SLE curves, which is based on the coupling between SLE and Liouville quantum gravity (LQG). Compared to prior methods that compute SLE related quantities via its coupling with LQG, the crucial new input is the exact solvability of structure constants in Liouville conformal field theory. It appears that various percolation exponents can be expressed in terms of conformal radii that can be computed this way. This includes known exponents such as the one-arm and polychromatic two-arm exponents, as well as the backbone exponents, which is unknown previously. In this talk we will review this method using the derivation of the backbone exponent as an example, based on a joint work with Nolin, Qian, and Sun.