200 EARTHWORK 



and the prismoidal correction is 



The true ^'olixme of the triangular prismoid is, therefore, 



V=V a + C 



A study of the correction will show that, if either the bases, 

 or the altitudes of the two end sections are equal, one of 

 the factors (bi bz) or (hi hi) will become zero, and therefore 

 the correction becomes zero. It shows also that, when one 

 or both of these factors are small, the correction is a corre- 

 spondingly small quantity; and that, when (as is usually the 

 case) the breadth and height at one section are both smaller 

 or both larger than the breadth and height at -the other sec- 

 tion, the correction is negative. Thus, if bz is less than bi 

 and hi is less than hi, then bi bi is positive, hz hi is negative, 

 and, therefore, C is negative. But when C is negative, V a 

 is greater than the true volume V; that is, the method of 

 averaging end areas usually gives a result that is too large. 

 When the difference of the breadths and heights is very large, 

 the correction is very large, and V a is very greatly in error. 

 Thus, for a pyramid, in which both bz and hz are zero, the cor- 



rection is / bihil 



-(6,-OHO-fc) 



The true volume is ^bihil, and therefore, the error in the 

 value of V a is one-half or 50%, of the true volume. This 

 extreme case shows the importance of computing the pris- 

 moidal correction when the areas of the bases are very unequal. 



EXAMPLE. The dimensions of the bases of a triangular pris- 

 moid are: &i = 18 ft., hi = 8 ft., fo=12 ft., and hz = 9 ft. Find 

 the volume of this prismoid, in cubic yards, if the length of 

 the prismoid is 100 ft. 



SOLUTION. The areas of the bases are: Ai=$X18X8 = 72 

 sq. ft., and Az=\ X 12 X 9 - 54 sq. ft. Substituting these 

 values in the preceding formula for V a , and dividing by 27 

 to reduce to cubic yards, 



y a = -HPx(72+54)-f- 27 = 233.33 cu. yd., nearly 



Substituting the given values in the formula for C, and divid- 

 ing by 27 to reduce to cubic yards, 



C = -WX(18-12)X(9-8)-4-27 = 1.85cu. yd. 





