# Calculating Surface area/volume from 2D cross section?

by SuperG
Tags: area or volume, cross, surface
 P: 5 I'm feeling a little stupid today and I need some help... Assume that I have a radially-symmetric 3D object (for example a candle stick or the base of a table lamp) and that I can calculate the surface area of the largest longitudinal cross section (ie, I split the object precisely along the axis of radial symmety and measure the area of the newly exposed surface). Also assume that the outside profile of this object can be described by a function (though I do not know what the function is) Is there any reason that I cannot then treat this object as the cross section of a simple cylinder, the product of whose length and diameter is equal to my object's cross sectional surface? In other words, if my object's cross sectional area turned out to be 10 square units with a length of 5), then can I say that my object's volume is equal to the volume of a cylinder with length 5 and diameter 2? EDIT: corrected an error
 P: 506 You're talking about a solid of revolution. In that case, take note of two different SORs, one cross-section has 2 unit circles each 1 unit symmetrically from the axis, and the other of 2 unit circles each 2 units symmetrically from the axis. The resulting torii have different volumes, but the same cross-sectional area. For objects which have the axis running through them, consider the unit circle and a square with sides of length $\sqrt{\pi}$. Then the volume of their SORs are $\frac{4\pi}{3}$ and $\frac{\pi^2\sqrt{\pi}}{2}$ even though their cross-sectional areas were the same. Also, for cross-sections of equal height and area, consider the absolute values of sine and cosine rotated.
P: 5
 Quote by hypermorphism You're talking about a solid of revolution. In that case, take note of two different SORs, one cross-section has 2 unit circles each 1 unit symmetrically from the axis, and the other of 2 unit circles each 2 units symmetrically from the axis. The resulting torii have different volumes, but the same cross-sectional area.
Assuming no hollow center though, would my method work?

I understand that what I'm assuming would not be universally applicable, but may rather be a special case.

I'm going to work on it some more to at least understand where this method fails (FWIW, this is not a class assignment but rather an applied problem--I'm trying to save my lab \$3000)