XXXll MENSURATION. 



3. Volumes. 



a. Volume of prism. 

 A = area of base, // :=: altitude, F= volume. 



F= A h. 



For an oblique triangular prism whose edges a, b, c are inclined at an angle a 



to the base, 



V^\ {a^h^c) A sin a. 



b. Volume of pyramid. 



A = area of base, h = altitude, F= volume. 



F= J A h. 



For a truncated pyramid whose parallel upper and lower bases have areas Ay 

 and A-i respectively and whose distance apart is //, 



V=\h {A. + y/^o ^1 + ^1). 



The volume of a wedge and obelisk may be expressed by means of the volumes 

 of pyramids and prisms. 



c. Volume of right circular cylinder. 

 r = radius of base, /i =^ altitude, F= volume. 



V= TT r- h. 



77 = 3.14159265, log 7r = 0.49714987. 



For an obliquely truncated cylinder (having a circular base) whose shortest and 

 longest elements are //j and Jio respectively, 



F=-Wr' (/a, + //,). 



For a hollow cylinder the radii of whose inner and outer surfaces are }\ and r.^ 

 respectively, and whose altitude is //, 



V=iT h{rl — r\) 



d. Volume of right cone with circular base. 



r =: radius of base, h = altitude, F=i volume. 



F= 1 TT r" h. 



For a right truncated cone the radii of whose upper and lower parallel bases 

 are Vy and r, respectively, and whose altitude is //, 



e. Volume of sphere and spherical segments, 

 r = radius of sphere, // = altitude of segment, F=: volume. 



