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Some Volumes

Some general principles: Suppose tex2html_wrap_inline1403 and tex2html_wrap_inline1405 are two similar figures. Assume that the ratio of two corresponding one-dimensional lengths is a/b. Then the ratio of two corresponding two-dimensional areas is tex2html_wrap_inline1409 , and the ratio of two corresponding three-dimensional volumes is tex2html_wrap_inline1411 .

The volume of a prism with base of area B and height h is Bh (area of base times height).

The volume of a pyramid with base of area A and height h is tex2html_wrap_inline1423 . You can prove this with calculus by placing the pyramid with apex at the origin and with base perpendicular to the z-axis. Then we can calculate the volume by making slices perpendicular to the z-axis. Let A(z) be the area of the slice at position z. Then tex2html_wrap_inline1433 , so tex2html_wrap_inline1435 . We get the volume by integrating from z=0 to z=h:


Carl Lee
Wed Apr 21 08:17:28 EDT 1999