We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical numerical radius of a bounded linear operator which improve on the existing ones. In particular, we prove that for a bounded linear operator A, 14‖A∗A+AA∗‖+μ2max{‖ℜ(A)‖,‖ℑ(A)‖}≤w2(A)≤w2(|ℜ(A)|+i|ℑ(A)|),where μ= | ‖ ℜ(A) + ℑ(A) ‖ – ‖ ℜ(A) – ℑ(A) ‖ |. This improve the existing upper and lower bounds of the numerical radius, namely, 14‖A∗A+AA∗‖≤w2(A)≤12‖A∗A+AA∗‖.