Ex. 9. Required the folidity of a cylinder, the diameter of its bafe being 15 inches, and length 14 feet.

Anf. 17.180625 cubic feet.

Ex. 10. What is the folidity of a pillar 60 inches diameter, Anf. 1099.56 cubic feet.

and 56 feet high?

PROBLEM V.

To find the fuperficies of any pyramid or cone.

RULE.

Multiply the primeter of the base by one half of the slant altitude, to the product add the area of the bafe, the fum will be the fuperficies.

The reafon of this rule is obvious: For if the base of the pyramid be any reAilineal figure, each of the fides will be triangles, whofe altitude is the fame with the flant altitude of the pyramid.

It is alfo plain, that the convex surface of a cone is the fector of a circle, whofe radius is the flant altitude, and arch the circumference of the cone's base.

EXAMPLE I.

Required the fuperficies of a right cone, whofe diameter of its bafe is 10 feet, and flant altitude 36 feet.

179

Ex. 2. Required the furface of a fquare pyramid, the fide of the bafe being 30 inches, and flant altitude 6 feet.

Anf. 36 fq. feet. Ex. 3. If the fide of the pentagonal bafe be 10 inches, and the flant altitude 5 feet, required the furface of the pyramid. Anf. 11.6114 ft feet. Ex. 4. What is the fuperficies of a hexagonal pyramid, whofe fide is 15 inches, and flant altitude 4 feet?

Anf. 19.0594875 Sq. feet.

PROBLEM VI.

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

RULE.

Multiply the area of the base by, the perpendicular altitude, and the product will be the folidity.

Note. Any pyramid is the third part of a prifm of the fame bafe and altitude : Also a cone is equal to one-third the circumfcribing cylinder.

EXAMPLE I.

Required the folidity of a pentagonal pyramid, whose perpendicular altitude is 60, and fide 8 feet.

Ex. 2. What is the folidity of a cone, whofe flant altitude is 96 inches, and diameter of its bafe 20 inches?

Anf. 9998.45616 cubic inches.

Ex. 3. Required the folidity of a cone, whofe perpendicular height is 5 feet, and diameter of its base 16 inches.

Anf. 2.3271 cubic feet.

Ex. 4. Required the folidity of a triangular pyramid, its height being 14 feet, and the three fides of its base 12, 14, 10 feet. Anf. 284.13716

PROBLEM VII.

To find the fuperficies of the fruftum of a cone, or any pyramid,

RULE.

Add together the primeter of both ends, and multiply one half the fum by the flant altitude, to the product add the area of both ends, and the fum will be the fuperficies.

EXAMPLE I.

Required the furface of the fruftum of a fquare pyramid, the fides of the leffer and greater ends being 14, and 24 inches, and dant altitude 2 feet 3 inches.

14X

14X4 56 the perimeter of the leffer end. 24×4 96 the perimeter of the greater end.

Ex. 2. Required the surface of the fruftum of a cone, the diameter at the greater end being 10, at the leffer 6 feet, and flant altitude 15 feet. Anf. 496.3728 fq. feet.

Ex. 3. What is the furface of the fruftum of a pentagonal pyramid, its flant altitude being 140 inches, and the fides of the ends 20, and 30 inches? Anf. 137.0598 fq feet.

PROBLEM VIII.

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

RULE

RULE I. *

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

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

RULE 3. To three times the fquare of half the fum of the two diameters, add the fquare of half their difference; multiply the fum by the height, and this product again by .7854, the last product will be the folidity.

RULE 4. Add the fquares of the two diameters to their product; this sum, multiplied by .7854, and again by the height, will give the folidity.

EXAMPLE I.

Required the folidity of the fruftum of a cone, whofe height is 20 inches, the greater diameter 32, and leffer 24 inches,

By RULE I.

32

32

1024

-7854

4096

5120

8192

7168

S04.2496 area of the greater bafe.

24

* Fhe fruftum of any pyramid or cone is equal to three complete pyramids