Was Jesus crucified on the same cross he had carried to Calvary?

Some traditional drawings of crucifixion of Jesus show him hanging on the cross with his feet at a man's height above the ground. That seems justified, as we read in Jn 19:28-29: > After this, when Jesus knew that all was now finished, he said (in order to fulfill the scripture), “I am thirsty.” A jar full of sour wine was standing there. So they put a sponge full of the wine **on a branch of hyssop and held it to his mouth.** Given that the convict was prone to give violent and painful jerks on the cross , it had to be positioned in a pre-dug hole say, of three feet on the ground. Let us presume that the vertical beam of Jesus' cross measured 12 feet, considering the portion that went to the ground, the position of his feet above ground and the top portion of vertical beam where INRI was placed. Now, Jesus was made to carry the entire cross, with the lowest end dragging on the path. In terms of geometry, the ideal length of the cross would be proportionate to his height so as to allow enough space between the shoulder and the vertical and horizontal beams built at 90 degree angle. The cross on which he was crucified appears too long for such a proportion. One is therefore, inclined to conclude that Jesus in fact carried a cross proportional to his height, and was crucified on a different cross vertically much longer. My question therefore is: Was Jesus crucified on the same cross he had carried to Calvary? Inputs from any denomination are welcome.