Math CentralQuandaries & Queries


Question from Tom, a teacher:

I am a ceramic teacher and wanted my students to make a 12 oz. cup, what formula should we use?


My favorite coffee mug (cylindrical outside diameter 3.25 in = 8.3 cm, height 3.75 in = 9.5 cm) holds 12 oz when filled to the brim. My advice is to go to a store that features a wide variety of cups and mugs, and choose the size and shape you prefer.

For the record, to answer your question you need three formulas:
To convert from fluid oz to cubic inches (in US units) use 1 oz = 1.8047 cu in. (so that 12 oz is about 21.66 cu in.)

For a cylinder, the volume equals pi*(height)*(the square of the radius) -- where pi equals about 3.1416. So for my cup, the inside radius is just under 1.5 in, while the inside height is about 3.3 in., which gives about 21.5 cu in, as desired.

For a cone, the volume is 1/3*pi*(height)*(the square of the radius). For a truncated cone you cut a small cone from the top of a larger cone, so the volume is (large cone) - (small cone). Thus you need to use the radius and the height of both the large and the small cone.

If the radii of the top and bottom disks are R and r, then the volume is
1/3*pi*(height)*(R2 + rR + r2)



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