Knowing the volume of a mountain, hill, volcano or any kind of elevation, the principle is based on finding the difference of two digital elevation models, the first with the actual topography of the land (with mountain) and the second by removing the geomorphological element (no mountain).
In this illustration a first TIN is created with real information of the contour lines, the second TIN builds eliminating the contours of geomorphological element (also by placing the same value to the contours of the mountain, in order to a flat surface in the area of geomorphological element), now just enough to get the volume difference between digital elevation models with the Surface Difference tool (TIN Difference in ArcGIS 9.x).
ArcToolbox > 3D Analyst Tools > Triangulated Surface > Surface Difference
ArcToolbox > 3D Analyst Tools > Terrain and TIN Surface > Surface Difference
The result is shown in a vector file, exploring the attribute table the difference in the volume of both digital elevation models is observed, in this illustration the mountain has a volume of about 900 million cubic meters.