MENSURATION AND SURVEYING 21 



Solid cone or pyramid — 

 Surface of cone = area of base + circumference X h 



the slant height. 

 Surface of pyramid = area of the base + areas of the 



side triangles. J 



Solid content = area of the base X one third of the 

 perpendicular height. 

 Sphere — 

 Surface = diameter 2 X 3' 14159 

 Solid content = diameter^ X '6236 

 Wedge — 

 Solid content = area of base X £ perpendicular height if 

 all the edges are equal. 



A Priamoid is a solid whose ends are parallel but unequal 

 in area, e.g., a bund whose extreme heights are unequal, a 

 cutting whose extreme depths are different, a gravel heap on 

 the roadside, a pond, etc. 



NOTE.— A prismoid includes a cone and a pyramid : but if the ends 

 besides being parallel are also equal and similar so that the solid is uni- 

 form from end to end it is a prism. 



Solid contents of a prism = area of one of the parallel ends 



X height or depth or length. 

 The solid contents of a prismoid may be calculated in 

 three ways — 



(a) Average of extreme areas X length (or height). 



(b) The middle area X length (or height). 



(c) By the following formula :— 



' h = distance between the parallel ends. 

 A x = area of one end. 

 A, = area of the other and parallel end. 

 Am = area of a section taken midway between Ai 

 and A2 and parallel to them. 



Solid contents A x ( Ai -f 4 Am + A2) 

 b 



(a) Always gives more than the real volume and may be 

 used for making estimates, so as to be on the safe side. 



(b) Always gives less than the real volume and is used 

 for measuring heaps of road metal stored by contractor . 



Example. — A metal heap has a rectangular base 14' X 6' 

 and a rectangular top surface 10' X 2' with a height of 2'. 

 Find its volume. 



(a) 84 + 20 X 2 = 104 cubic feet. 



(b) 12 X 4 X 2 = 96 cubic feet. 



(0) 84 + 4 X 48 + 20 x 2 = 9g . 6 cubic f ^ 



