In this paper, we present arc-length estimations for quadratic rational B$acute{e}$zier curves using the length of polygon and distance between both end points. Our arc-length estimations coincide with the arc-length of the quadratic rational B$acute{e}$zier curve exactly when the weight ${omega}$ is 0, 1 and ${infty}$. We show that for all ${omega}$ > 0 our estimations are strictly increasing with respect to ${omega}$. Moreover, we find the parameter ${mu}^*$ which makes our estimation coincide with the arc-length of the quadratic rational B$acute{e}$zier curve when it is a circular arc too. We also show that ${mu}^*$ has a special limit, which is used for optimal estimation. We present some numerical examples, and the numerical results illustrates that the estimation with the limit value of ${mu}^*$ is an optimal estimation.