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14*4=56 the perimeter of the lefser end. 24X4=96 the perimeter of the greater end.

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Ex. 2. Required the surface of the frustum of a cone, the die ameter at the greater end being 10, at the lesser 6 feet, and fant altitude 15. feet.

Anf. 496.3726 sq. feet. Ex. 3. What is the surface of the frustum of a pentagonal pyramid, its flant altitude being 140 inches, and the sides of the ends 20, and 30 inches?

Anf. 137.0598 /afeet.

PROBLEM VIII.

To find the folidity of the fruftum of a cone, or any pyramid.

RULE RULE I. *

Add into one sum the area of both ends, and the inean proportional between them; multiply the sum by the perpendicular height, and the product will be the folidity.—This rule ferves whether the frustum be of a cone or pyramid. The three following applies to the frustum of a cone.

RULE 2. To the product of the two diameters add the square of their difference, multiply the sum by the height, and this product again by .7854 for the folidity.

Rule 3. To three times the square of half the sum of the two diameters, add the square of half their difference ; multiply the sum by the height, and this product again by :7854, the last product will be the solidity.

Rule 4. Add the squares of the two diameters to their product; this sum, multiplied by .7854, and again by the height, will give the solidity.

EXAMPLE I.

Required the solidity of the frustum of a cone, whose height is 20 inches, the greater diameter 32, and lesser 24 inches.

By RULE I.

32

32

1024 .7854

4096 5120 8192 7168

$04.2496 area of the greater base.

24

Fhe frustum of any ryramid or cone is equal to three complete pyramids

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24
24

96 48

576 7854

2304 2880 4608 4032

452.3904 area of the lefler bafc.

804.2496 X 452.3904=363834-79824384 And
363834.798243843603.1872 mean proportional.

804.2496 603.1872 452.3904 1859.8272 fum.

20 height. 3)37196.5440

12398.8480 folidity.

By

or cones of the same altitude with the frustum, whereof the greatest of the three has its bafe equal to the greater base of the frustum; the least has its base equal to the less base of the frustum, and the other a mean proportional bos tween them,

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