# FRUSTUM

, in Geometry, is the part of a solid next the base, left by cutting off the top, or segment, by a plane parallel to the base: as the Frustum of a pyramid, of a cone, of a conoid, of a spheroid, or of a sphere, which is any part comprised between two parallel circular sections; and the Middle Frustum of a sphere, is that whose ends are equal circles, having the centre of the sphere in the middle of it, and equally distant from both ends.

*For the Solid Content of the Frustum of a cone, or of
any pyramid, whatever figure the base may have.* Add
into one sum, the areas of the two ends and the mean
proportional between them; then 1/3 of that sum will be
a mean area, or the area of an equal prism, of the same
altitude with the Frustum; and consequently that
mean area being multiplied by the height of the Frustum,
the product will be the solid content of it.
That is, if A denote the area of the greater end,
*a* that of the less, and *h* the height;
then is the solidity.

Other rules for pyramidal or conic Frustums may be seen in my Mensuration, p. 189, 2d edit. 1788.

*The curve Surface of the Zone or Frustum of a sphere,*
is had by multiplying the circumference of the sphere
by the height of the Frustum. Mensur. p. 197.

*And the Solidity of the same Frustum is found,* by adding
together the squares of the radii of the two ends.
and 1/3 of the square of the height of the Frustum, then
multiplying the sum by the said height and by the
number 1.5708. That is, is
the solid content of the spheric Frustum, whose height
is *h,* and the radii of its ends R and *r,* *p* being =
3.1416. Mensur. p. 209.

For the Frustums of spheroids, and conoids, either parabolic or hyperbolic, see Mensur. p. 326, 328, 332, 382, 435. And in p. 486 &c, are general theorem<*> concerning the Frustum of a sphere, cone, spheroid, or| conoid, terminated by parallel planes, when compared with a cylinder of the same altitude, on a base equal to the middle section of the Frustum made by a parallel plane. The difference between the Frustum and the cylinder is always the same quantity, in different parts of the same, or of similar solids, or whatever the magnitude of the two parallel ends may be; the inclination of those ends to the axis, and the altitude of the Frustum being given; and the said constant difference is 1/4 part of a cone of the same altitude with the Frustum, and the radius of its base is to that altitude, as the fixed axis is to the revolving axis of the Frustum. Thus, if BEC be any conic section, or a right line, or a circle, whose axis, or a part of it, is AD; AB and CD the extreme ordinates, FE the middle ordinate, AF being = FD; then taking, as AD to DK, so is the whole fixed axis, of which AD is a part, to its conjugate axis; and completing the parallelogram AGHD: then if the whole figure revolve about the axis AD, the line BEC will generate the Frustum of the cone or conoid, according as it is a right line or a conic section, or it will generate the whole solid when AB vanishes, or A and B meet in the same point; likewise AGHD will generate a cylinder, and ADK a cone: then is the 4th part of this cone always equal to the difference between the said cylinder generated by AGHD and the solid or Frustum generated by ABECD; having all the same altitude or axis AD.

In the parabolic conoid, this difference and the cone vanish, and the Frustum, or whole conoid ABECD, is always equal to the cylinder AGHD, of the same altitude.

In the sphere, or spheroid, the Frustum ABECD
is *less* than the cylinder AGHD, by 1/4 of the cone
AKD. And

In the cone or hyperboloid, that Frustum is *greater*
than the cylinder, by 1/4 of the said cone AKD, which
is similar to the other cone IBCD.

It may be observed, that the same relations are true, whether the ends of the Frustum are perpendicular or oblique to the axis. And the same will hold for the Frustum of any pyramid, whether right or oblique; and such a Frustum of a pyramid will exceed the prism, of the same altitude, and upon the middle section of the Frustum, by 1/4 of the same cone.

It has been observed, that the difference, or 1/4 of the cone AKD, is the same, or constant, when the altitude and inclination of the ends of the Frustum remain the same. But when the inclination of the ends varies, the altitude being constant; then the said difference varies so as to be always reciprocally as the cube of the conjugate to the diameter AD. And when both the altitude and inclination of the ends vary, the differential cone is as the cube of the altitude directly, and the cube of the said conjugate diameter reciprocally: but if they vary so, as that the altitude is always reciprocally as that diameter, then the difference is a constant quantity.

*Another general theorem for Frustums, is this.* 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 4 times the middle fection, 1/<*> of the
last sum will be a mean area, and being drawn into the
altitude of the solid, will produce the content. That
is, is the content of
ABCD.

And this theorem is general for all Frustums, as well as the complete solids, whether right or oblique to the axis, and not only of the solids generated from the circle or conic sections, but also of all pyramids, cones, and in short of any solid whose parallel sections are similar figures.

The same theorem also holds good for any parabolic
*area* ABECD, and is very *nearly true* for the area of
any other curve whatever, or for the content of any
other solid than those above mentioned.