E (6.) Every Prism on a triangular Base can be divided into three equal Pyramids. Let ABE, DFC be the prism; draw a plane through DE and E, cutting off the pyramid E, DFC, and through ED and B draw a plane dividing the remainder into two pyramids E, ABD and E, BCD. Now since these two latter have their vertexes coincident at E, and since their bases ABD, BDE are manifestly equal being halves of the parallelogram AC, the two pyramids E, ABD and E, BDE are equal. Now the pyramid E, ABD is clearly the same as the pyramid D, ABE. But the base ABE = base DEF, and the perpendicular from E on DFE, is equal to that from D on ABE, since these planes are parallel; hence the pyramid D, ABE = pyramid E, DFE, and therefore the three pyramids are equal. COR. 1.-Hence if a pyramid and a prism have equal triangular bases and are of equal altitudes, the pyramid is one third part of the prism. Fig. 38. COR. 2.-Let A area of base, h= height of pyramid. Then the volume of the Ah prism on base A and of height h = Ah. And .. the volume of pyramid = 3 . so that COR. 3.-If we have a pyramid on a polygonal base of which the area is A, it can be divided into triangles whose areas we will suppose to be A, A2 Aŋ.. A = A, + A2 + A ̧ + Now if h be the height of the pyramid, then its volume being clearly the sum of the pyramids whose bases are A, A, Ag.... and height h, will equal ... COR. 4.-This is true, however great the number of sides 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, and the limit of the pyramid is the circumscribing Ah cone .. volume of cone = where A is the area of the circular base. F D E (7.) To determine the Volume of the Frustum of a right Prism on a triangular Base. Let ABC, DEF be the frustum, where ABC is perpendicular to the edges. Let A = area of ABC, and let h, h2 h, be the edges AF, BE, CD respectively, and V the required volume. Join FC and suppose plane to pass through FCE, cutting off the pyramid C, FED, and another through ECA cutting off the pyramids E, ABC, E, ACF, these three make up the volume V. Join FB, DB, and DA. Now volume of h E, ABC = A ; volume of E, FAC volume of 3 B, FAC; since perpendiculars from E and B on the plane ACF are clearly equal, and B, FAC is the same h pyramid as F, ABC, the volume of which is A 3 Again, since FA is parallel to DC, the triangle ACD B Fig. 39. is the triangle FCD, and we have already seen that the perpendiculars from E and B on the plane ACDF are equal, .. the pyramid E, CDF = the pyramid B, ACD, Cor: If the prism be ABC FED, in which neither of the ends is perpendicular to the edges, take abc an area whose plane is perpendicular to the edges, and let A = area of abc. XC + x2 + X3 + 1 x1 + Y + X 2 + Y 2 + x3 + Yz A = A (8.) To determine the Volume of a Frustum of a Right Prism on a base which Let ABCDEFGH be the frustum in question; let h, h2 h2 h be the edges at A. B. C. D. respectively; draw AC, and if a plane E, pass through AC and E it divides AB. . . . 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 If we had divided the figure by a plane passing through BD, we should have had Similarly if we had divided at C and D, we should have had respectively C h2 + h2 + h ̧. Then adding the four values of V we have 4 V = A, ( S + S + S ) + 4 (S + S + S ) (9). To find the Volume of a Frustum of a right Prism on a regular Pentagonal Base. 2 3 3 Let P, P, P, P, P, be the base, and let h h h h h, be the edges corresponding to these angles. Join P, P, and P1 P1, and let the areas 1 4 3 4 3 1 of P, P, P, and P1 P, P., which are equal, be each A1, and the area of P, P, P, be 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 Now divide the base by lines drawn through P2 the areas will be the same as before, and hence Fig. 42. A2 ( + h ̧ + h2) + A1 (h2 + b ̧ + kg) 5 3 1 3 A2 Then adding these together and writing we have 5 V = The student can easily prove that a similar formula is true in the case of a prism on a hexagonal base, or indeed on any base which is a regular polygon. BCNM perpendicular to the plane of the rectangle, these divide the given figure into three, viz., AEKDLG, BDN, and BMFCHN. Let AB = α AD = = h AK = x DL = x, EK b GL = 6, MF = c NH = c, Then (a) the figure BDN is of the same kind as in article 8. Now the area ᎠᏴ = ah, if therefore V, is its volume, 1 (b.) Let V, be the volume of the figure AEKDLG, draw planes through ED,K and GD,K, dividing the figure into three pyramids, DGL,K, AEK,D, and DEG,K. Now the volume of GED,K is to that of ADE,K as GED is to ADE, or as AE to GD, or since triangles AEK, GDL are similar, these volumes are as AK to DL, or as EK to GL. D is clearly four times the area of a section made by a plane parallel to ABFE and DEHG, and half way between them. Hence the solid content of a prismoid is found by the following rule: "To the areas of the ends add four times the area of the mean section, multiply this sum by 4th of the height of the figure; this product is the volume required." = In practical cases it will generally happen that b = c, and b, c,. This does not affect the enunciation of the rule, but simplifies the formula, which becomes (11.) To find the Solid Content of a Railway Cutting. In the last article the figure is very nearly that of a portion of a railway cutting, in which ABCD is the road, AG and BH, the sloping sides of the embankment, and hence the solid content required can be found by means of the rule given in the last article. It is to be observed that the rule requires E G and FH to be straight lines, cr, as an approximation, to be very nearly straight lines, or EH to be a plane, which is not true if the cutting is a long one. For this case we derive the following rule from the above formula. Let ABC represent a section made by a vertical plane of the hill to be cut through, AB the level of the road, and suppose sections of the cutting to be made by planes perpendicular to AB, at equal distances along that line, viz., at M, M, M, ... M2n+1, let the terminal sections at A and B be a and b, and the sections at P,M, P.M. P1 P2 P3... Pan P2n+1, the number of sections being odd, and let the common distance between the sections be h. Now by last article the volume of portion h and the volume of the portion P, M, MP, is (P + 4 P3 + P1), and so on, h and the volume of Pan Man BC is (P2 n + 4 P2 +1 + b). 3* |