FRUSTUM, in mathematics, a part of some solid body separated from the rest.
The frustum of a cone is the part that remains, when the top is cut off by a plane parallel to the base ; and is other wise called a truncated cone. The frus tum of a pyramid is also what remains, after the top is cut off by a plane parallel to its base. To find the solid content of the frustum of a cone, pyramid, &c. the base being of any figure whatever : add the areas of the two ends and the mean proportional between them together, then one-third of that sum will be the mean area, or the area of an equal prism, of the same altitude with the frustum ; and con sequently, that mean area multiplied by the height of the frustum will give the so. lid content for the product : If A = area of the greater end a lesser end = height : then A a — X /a = the solidity.
3 The frustum of a globe or sphere is any part thereof cut off by a plane, the solid contents of which may be found by this rule. To three times the square of the semi-diameter of the base, add the square of its height ; then multiplying that sum by the height, and this product multiplied by 5236, gives the solidity of the frustum. A frustum, or portion of any solid, generated by the revolution of any conic section upon its axis, and ter minated by any two parallel planes, may be thus compared to a cylinder of the same altitude, and whose base is equal to the middle section of the frustum made by a parallel plane. 1. The differ ence between such frustum and cylinder is always the same in clifferentparts of the same or of similar solids, when the in clination of the planes to the axis andthe altitude of the frustum are given. 2. In
the parabolic conoid„ this difference va nishes ; the frustum being always equal to a cylinder of the same height, upon the section of the conoid that bisects the altitude of the frustum, and is parallel to its bases. 3. In the sphere, the frustum is always less than the cylinder, by one fourth part of a right angled cone of the same height with the frustum ; or by one half of a sphere, of a diameter equal to that height : and this difference is always the same in all spheres whatever, when the frustum the frustum t timal always exceeds In the he cylinder, by one fourth part of the content of a similar cone, that has the same height with the frustum.
As a general theorem : in the frustum of any solid, generated by the revolution of any conic section about its axis : if to the sum of the two ends be added four. times the middle section, then the last sum divided by six will he the mean area, and being drawn into the altitude of the solid will produce the content : That is, A and a being the areas of the ends, M . equal the middle section, then we have A h= solid content.
6 This theorem holds good for complete solids as well as frustums, whether right or oblique, and not only of the solids ge nerated from the conic sections, but also of all pyramids, cones, audio short of any solid, whose parallel sections are similar figures.