The volume of a frustum

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Question from dave:

I have a frustum top 1.7r bottom .55r and 2.14 h
I have to calculate cement in a silo every week.
I am not very good at maths but i have been adding the top and bottom
to get an average so as to turn it into a cylinder and i come up with
8.5m3 I know that the correct volume is 9.24 m3. Can you tell me why
getting an average width on the top and bottom of a frustum doesnt work.
Thank you.

Dave,

Your mistaken idea of using an average probably comes from finding the area of a trapezoid (namely Area = height of the trapezoid times the average of its parallel sides). Of course, the cross section of a frustum by a plane through its axis is indeed a trapezoid, but you need a bit of calculus to go from there to the volume. You can find the volume easily without calculus. See
http://en.wikipedia.org/wiki/Frustum/

The correct formula for the volume of a frustum can be found on that page or in our mathcentral files,

http://mathcentral.uregina.ca/QQ/database/QQ.09.07/s/gina1.html

These formulas tell you that the procedure is more complicated than finding the simple average of the top and bottom areas.

Chris

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.