on 



MATHEMATICS. MEXSm AT ION. 



[VOU'MKS Of PRISMS. 



.. 1. Hence if a pyramid and a prism have equal 

 triangular bite* and are of equal altitude*, the pyramid 

 i* one- third part of the prism. 



-Let A area of hue, A hoightof pyramid. 

 Then the volume of the prism on base A and of height 



A AA. And /. the volume of pyramid - '- 







i;. 3. If we have a pyramid on a polygonal base of 

 which the area is A, it can be divided into triangle* 

 whone areas we will suppose to be A, A 2 A s . . . . so that 

 A - A, + A, + A, + . . . Now if A be the height of 

 the pyramid, "then 'it* volume being clearly the sum of 

 the pyramids whose bases are A t A a A 8 . . . . and height 

 A, will equal 



+ . .\~l\h. 



COB. 4. This is true, however great the number of 

 side* there are to the polygon, and hence is true in the 

 limit Now if we suppose the polygon to be regular, its 

 limit is the circumscribing circle, aud the limit of the 

 pyramid is the circumscribing cone .'. volume of cone = 



- where A is the area of the circular base. 

 3 



(T). To determine the Volume of the Frustum of a right 

 Prism on a triangular BOM. 



Let ABC, DEFbethefrus- 

 turn, where A B C is perpen- 

 dicular to the edges. Let A = 

 area of ABC, and let h. h., h s 

 be the edges A F, B E, CD re- 

 spectively, and V the required 

 volume. Join FC and suppose 

 a plane to pass through FOE, 

 cutting off the pyramid C, FED, 

 and another through E C A cut- 

 ting off the pyramids E, ABC, 

 E, ACF, these three make up 

 the volume V. Join F B, D B, 

 aud D A. Now volume of E, 



Kg. M. 



A B C A 



volume of E, 



F A C = volume of B, F A C ; since perpendiculars from 

 E and B on the plane ACF are clearly equal, aud B, 

 F A C is the same pyramid as F, A B C, the volume of 



which is A '.'. 







Again, since F A is parallel to D C, the triangle ACT) 

 is = tlie triangle F C D ; and wo have already seen that 

 the perpendiculars from E aud B on the plane A C D F 

 are equal, .'.the pyramid E, CDF = the pyramid B, 



A C D, i. e. = D A B C, the volume of which is 



- 

 3 



' .0. Q T " .. Q " 



o O o 



COR. If the prism be A B C, F R D, 

 in which neither of the ends is perpen- 

 dicular to the edges, take al>c an area 

 whose plane is perpendicular to the 

 edges, and let A = area of abe. 



Let A E = A, B D = A 2 C F = A 3 



oA = y, 6B y s cC = y 8 



Then if V H volume required 

 V~a6c. DKF + a&eABO 



-1 



s . l 



- A 



(8). To determine the Volume of a Fntstum of a 

 Prism on a bate which it a Parallelogram. 

 Let A B C D E F O H be Fig. . 



the frustum in question ; 

 let /i, h, A. A 4 bo the edge* 

 at A. B. C.D. respectively ; 

 draw AC, and if a plane 

 pass through A C and K it 

 divides A B .... H into 

 two frustums similar to 

 that in the last proposition. 

 Let V be the required 

 volume, and A, the area 

 ABC which is area ADC 



(Euclid, I., 34). Now the volumes of the two triangular 

 prisms are respectively 



A, 



.-. v = 



A, 



If we had divided the figure by a plane passing through 

 B D, we should have had 



^ o" A! (A 2 + A s + A^) + 5-A, (A^ -f- A, + A 2 



Similarly if we had divided at C and D, we should have 

 had respectively 



V= =- A, (7i 3 -(- A ^ -\- A t ) -|- A, (A, -f- A 2 -f- A s ) 



V= A. (hi + h, 4- An) + ^r- A. (hn + /In + A.) 



3 ' v * T l *' 3 



Now let S = A, + A a + A s + A 4 . Then adding the 

 four values of V we have 



1 



/.If A be the whole area A B C D. Since A = 2A, 



V - A - A ~- > " 



4 4~ 4 



(9). To find the Volume of a Frustum of aright Pritm on 



a regular Pentagonal Base. 



Let P, Pj P, P* P, be the base, and let h^ h, h, h t 

 hi be the edges corresponding 

 to these angles. Join P t Pa 

 and P, P 4 , and let the areas 

 of P, P, P s and P, P P., 

 which are equal, be each Aj, and 

 the area of P t P, P 4 be A a . 

 Let V be the required volume, 

 then this volume consists of three 

 portions of prisms of the same 

 kind as in article (7) and hence 



V = - (h, 



A,) 



-L ^ (h -(-ft + h ) 4- ^ l (h -t-h 4- h\ 

 3 3 ^^ 



Now divide the base by lines drawn through P a the 

 areas will be the same as before, and hence 



h h, + ht ) + A *- (h t + h t +h) 



+ A ' <*. + A a + A,) 



and similarly by dividing the base by lines drawn suc- 

 cessively through Pa P and P, wo obtain 



V - If 



l (h, 



A 4 



- f (A., 



+ M 



A, 



V - -^ (h, +h,+ A,) + *'- (A. -f A, -f h,) 



+ -q (A* + A, + h 3 ) 



