890
THEORY AND PRACTICE OF ENGINEERING.
Book II.
To find either the Surface or solid Content of any of the regular Bodies , multiply thetabular area, taken from the following table, by the square of the linear edge of the solid,for the superficial content; and for the solid content, multiply the tabular solidity by thecube of the linear edge.
Number ofSides.
Name.
Surface.
Solidity.
4
Tetraedron -
1 *7320508
0*1178513
6
Hexaedron -
6*0000000
1-0000000
8
Octaedron -
3-4641016
0-4714045
12
Dodecaedron
20*6457288
7-6631189
20
Icosaedron -
8 -6602540
2-1816950
The diameter of a sphere being given, to find the side of any of the Platonic bodiesthat may be either inscribed on the sphere, or circumscribed about the sphere, or that isequal to the sphere.
Multiply the given diameter of the sphere by the proper or corresponding number in thefollowing table, and the product will be the side of the Platonic body required: —
The diameter of a sphere being
1, the side of a
That may be in-scribed in the sphereis,
That maybe circum-scribed about thesphere is,
That equal to thesphere is.
Tetraedron -
0*816497
2*44948
1*64417
Hexaedron -
0*577350
1*00000
0*88610
Octaedron -
0*707107
1 *22474
1*03576
Dodecaedron ~
0*525731
0-66158
0-62153
Icosaedron -
0-356822
0*44903
0*40883
Fig. 1222.
To find the Solid Content of a Sphere. — Multiply the cube of thediameter by *5236, and the product will be the solidity.
Or we may put d = the diameter, c = the circumference, and s = thesurface of the sphere or its circumscribing cylinder; also a = 3*1416,then ] 5 = the base of the cylinder, or one great circle of the sphere,and d is the height of the cylinder; therefore \ d s is the content of thecylinder : but § of the cylinder is the sphere ; that is § of \ ds, or £ dsis the sphere.
Again, because the surface s=arf s , therefore s = ^ad a = 5236 d 3the content; also d being = c -j- o, therefore £ad s = £c 3 -ra* = *01688.
Then if we cube the diameter of a globe, and multiply it by *5236, orcube the circumference and multiply it by *01688, we shall obtain thesolid content.
To find the Solidity of a spherical Segment , or Plano-convex Portion
of a Sphere _To three times the square of the radius of the base or
flat side, add the square of the versed sine or height; then multiplythe sum by the height, and the product so obtained by *5236 for thesolid content; or, from three times the diameter of the sphere takedouble the height of the segment; then multiply the remainder by thesquare of the height, and the product by the decimal *5236 for thecontent.
To find the Content of a solid Ellipse or Spheroid. — Multiply thesquare of the transverse by the square of the conjugate diameter, andthe product by *5236.
To find the Solidity of a Parabolic Conoid. — Multiply the square ofthe diameter of the base by the height or length of the axis, and theproduct by *3927: two such solids united at the base form a parabolicspindle.
To find the Content of a cylindrical Ring. — Add to the diameter ofthe cylinder of which the ring is formed the extent of the inner diameterof the ring; then multiply the sum by the square of the thickness on
the diameter of the ring, and the product by 2*4674, which is one fourth of the square of3*1416, and it will give the solidity.
To find the convex Surface of a Sphere. — Multiply the diameter of the sphere by itscircumference; or multiply 3*1416 by the square of the diameter, and the product will bethe convex surface required.
Fig. 1223.
Fig. 1224.