Cones and cylinders which are on equal bases are to one another as their heights.
等底圆柱（或圆锥）的体积之比等于它们的高之比。
For let EB, FD be cylinders on equal bases, the circles AB, CD; I say that, as the cylinder EB is to the cylinder FD, so is the axis GH to the axis KL. For let the axis KL be produced to the point N, let LN be made equal to the axis GH, and let the cylinder CM be conceived about LN as axis.
设圆柱EB和FD有相等底圆AB和CD，轴分别为GH和KL。延长KL至N使LN等于GH，并以LN为轴作圆柱CM。
Since then the cylinders EB, CM are of the same height, they are to one another as their bases. [XII. 11] But the bases are equal to one another; therefore the cylinders EB, CM are also equal.
因圆柱EB和CM等高，由XII.11，它们体积之比等于底面积之比；又底面积相等，故圆柱EB等于CM。
And, since the cylinder FM has been cut by the plane CD which is parallel to its opposite planes, therefore, as the cylinder CM is to the cylinder FD, so is the axis LN to the axis KL. [XII. 13] But the cylinder CM is equal to the cylinder EB, and the axis LN to the axis GH; therefore, as the cylinder EB is to the cylinder FD, so is the axis GH to the axis KL.
圆柱FM被平行于底面的平面CD所截，由XII.13，圆柱CM与FD之比等于轴LN与KL之比。
But, as the cylinder EB is to the cylinder FD, so is the cone ABG to the cone CDK.
因CM等于EB，LN等于GH，故EB与FD之比等于GH与KL之比；由XII.10，此比也等于圆锥ABG与CDK之比。