MATHS BITE: Steinmetz Solid

Named after mathematician Charles Proteus Steinmetz, a Steinmetz solid is the solid body generated by the intersection of two or three cylinders with the same radii at right angles.

Bicylinder_Steinmetz_solid
Bicylinder | Source: Wikipedia

These solids are named after Steinmetz as he managed to determine the volume of the intersection, though these solids were known long before he studied them.

If two/three cylinders intersect then the intersection is called a bicylinder/tricylinder.

Bicylinder

File:Steinmetz-solid.svg
Bicylinder | Source: Wikipedia

Surface Area: 16r2

Volume: {\frac  {16}{3}}r^{3}, where r is the radius of both cylinders.

The volume of the cube minus the volume of the eight pyramids is the volume of the bicylinder: the volume of 8 pyramids is \textstyle 8\times {\frac  {1}{3}}r^{2}\times r={\frac  {8}{3}}r^{3}.

 Then the bicylinder volume is:

\textstyle (2r)^{3}-{\frac  {8}{3}}r^{3}={\frac  {16}{3}}r^{3}

Tricylinder

Surface Area: 3(16-8{\sqrt  {2}})r^{2}.\,

Volume: (16-8{\sqrt  {2}})r^{3}\,

Tricylinder Steinmetz solid.gif
Tricylinder | Source: Wikipedia

 

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