A pair of free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other class. Given an orientable hyperbolic surface $F$ with negative Euler characteristic, any self intersecting geodesic $\alpha$ on $F$ and any self-intersection point $P$ of $\alpha$, we construct infinitely many pairs of length equivalent curves in $F$. We also relate these examples with the terms of Goldman bracket.