1250 the area of the sector AFCB. 8.75 X 30= 262.5 the area of the triangle ACF. 937.5 the area of the rev. segm. ACB. 49375 60125 29625 29500 4320.3125 Ex. 2. Required the folidity of a circular spindle, whose length is 30, and thickness 224 inches. Anf: 7350.853125. Ex. 3. Required the solidity of a circular spindle, whose middle diameter is 36, and length 40 inches. Anf. 29919 cubic inches. PROBLEM XVIIT. To find the folidity of the middle zone of a circular spindle. RULE. From the fourth part of the {quare of the length of the whole spindlę, subtract the square of half the length of the middle frustum, and multiply the remainder by the length of of the frustum : Multiply the central distance by the revolving area which generates the frustum ; then subtract this latter product from the former, and multiply the remainder by 3.1416, and twice the product will be the solidity. EXAMPLE I. Required the folidity of the frustum of a circular spindle, whose length is 40, greatest diameter 36, and least 16 inches. Draw EG parallel to mn, then EF shall be equal , mn, =20 and EF2 +FB-=EB=500 chord. EB2 500 -= 50 diameter of the generating circle. FB 10 Hence rad. BD = 25 AL-=AD2-LD-=625—495576 133.3 3 3 442.6 20 8853.3 first product. BE Ex. 2. Required the folidity of a circular spindle, whose length is 40, its greatest diameter 32, and least 24 inches. Anf. 27287 cubic inches. PROBLEM PROBLEM XIX. To find the superficies and folidity of the five regular or Platonic bodies. RULE. Multiply the square of the given fide into the corresponding tabular area for the superficies. And Multiply the cube of the given fide by the proper tabular solidity, for the solidity of the given body. Names. Containing fides. Area. Solidity. 1. Tetraedron 4 equilateral trian. 1.732051 0.117851 6 equal squares 6. 8 equal equi. lat. tri. 3.464102 0.471405 1 2 equal pentagons 20.645729 | 7.663119 20 equal equilat. tria. | 8.660254 | 2.181695 This table exhibits the area and folidity of any of the above bodies, the fide being unity. The areas of the above figures are so related to those of regular polygons, and their folidities to problems already treated of, that we shall leave the construction of the table for the exercise of the learner. EXAMPLE I. Fig. 97. Required the area and solidity of a tetraedron, whose fide is 30. |