2. What is the content of the middle frustum of a circular spindle, whose length is 20, greatest diameter 18, and least diameter 8?

Ans. 3657'1613.

PROBLEM XVIII.

To find the superficies or solidity of any regular body.

RULE.

1. Multiply the proper tabular area (taken from the following table) by the square of the linear edge of the solid, for the superficiesa

2. Multiply the tabular solidity by the cube of the linear edge, for the solid content.

EXAMPLES.

1. If the linear edge of a tetraedron be 3; required its surface and solidity.

tab. sur.

The square of 3 is 9, and the cube 27. Then,

3 1'73205

Oʻ11785 tab. sol. 5

27

Superficies. 15.58845

82495 23570

Solidity 3*18195

2. What is the superficies and solidity of a hexaedron, whose linear side is 2 ?

3. Required the superficies and solidity of an octaedron, whose linear side is 2.

4. What is the superficies and solidity of a dodecaedron,

a whose linear side is 2 ?

5. Required the superficies and solidity of an icosaedron, whose linear side is 2.

Ans.

Superficies 34:64100.
Solidity 17945352

PROBLEM XIX.

To find the surface of a cylindrical ring.

RULE.

This figure being only a cylinder bent round into a ring, its surface and solidity may be found as in the cylinder ; namely, by multiplying the axis, or length of the cylinder, by the circumference of the ring, or section, for the surface ; and by the area of a section, for the solidity:

Or

Or use the following rule for the surface :--To the thick, ness of the ring add the inner diameter ; multiply this sum by the thickness, and the product again by 9:8696, or the square of 3-1416.

EXAMPLES.

1. Required the superficies of a ring, whose thickness AB is 2 inches, and inner diameter BC is 12 inches.

2. What is the surface of a ring, whose inner diameter is 16, and thickness 4 ?

Ans. 789-568.

To the thickness of the ring add the inner diameter ; then multiply the sum by the square of the thickness, and

the

the product again by 2.4674, or of the square of 3'1416, for the solidity.

EXAMPLES.

1. Required the solidity of a ring, whose thickness is 2 inches, and its inner diameter 12.

2. What is the solidity of a cylindrical ring, whose thickness is 4, and inner diameter 16 ?

Ans. 789-568.

PROBLEM XXI,

To find the solidity of a spheroid,

RULE.

Square the revolving axis, multiply that square by the fixed axis, and multiply the product by .5236, for the content.