556 



CAMBRIA STEEL. 



SURFACES AND VOLUMES OF SOLIDS 



!* D *! 



CIRCULAR RING (TORUS) 



D & R=Mean Diameter and Mean Radius, 



respectively, of Ring 

 d & r = Mean Diameter and Mean Radius, 



respectively, of Section 

 Surface =7r 2 Dd =47r2Rr 



Volume * 



PRISMOID 



End faces are in parallel planes. 



Volume =-(A+A'+4M), where 



6 



1 = perpendicular distance between ends 

 A, A' =areas of ends 



M =area of mid section, parallel to ends 



UNQULAS FROM RIGHT CIRCULAR 

 CYLINDER 



(As formed by cutting plane oblique to base) 



I. Base, abc, less than semicircle; 

 Convex Surface 



=h(2re - (d Xlength arc abc))^- (r -d) 

 Volume=h(^e-(dXareabaseabc))-+-(r-d) 

 II. Base, abc, = semicircle; 



Convex Surface = 2rh Volume = %r 2 h 



III. Base, abc, greater than semicircle (figure); 

 Convex Surface 



= h(2re + (d Xlength arc abc))-^-(r+d) 

 Volume =h(He 3 + (d Xarea base abc))-r-(r+d) 



IV. Base, abc, =circle, oblique plane touching 

 circumference. 



Convex Surface = *rh Volume = H r 2 h 

 V. Base, abc,=circle, oblique plane entirely 

 above (figure). 

 Convex Surf ace = 2?rr 



X>i(h, minimum-f H, maximum) 

 Volume = u-r 2 X H (h, minimum 



+H, maximum) 



ANY SOLID OP REVOLUTION 



Let abed represent the generating section about 



axis A-A of solid abef . 

 Let g at distance h from A-A be the center of 



gravity of abed. 

 Let a be the angular amount of generating 



revolution. 

 Then 



Total Surface of solid abef 



= (27rha-7-360) X perimeter abed 



Volume of solid abef =( 2irha -4-360) Xarea abed 



For complete revolution(27rha-i-360) =2*rh 







