275. The Altitude of a cylinder is a straight line

joining the centres of its bases.

276. A Pyramid is a solid having for its base any polygon whatever, and for its lateral faces several triangles which meet in a common point called the vertex.

277. A Cone is a solid having a circular base, and its convex

surface tapering uniformly to a point called the vertex.

278. The Altitude of a pyramid or cone, is the perpendicular distance from its vertex to the plane of its base.

279. The Slant Height of a pyramid is the perpendicular distance from the vertex to one of the sides of the base; while that of a cone is measured by a

line drawn from the vertex to the circumference of the base.

280. When the top of a pyramid or cone is cut off by a plane parallel to the base, the part remaining is called the Frustum,

281. A Sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within called the Centre.

282. A straight line drawn. through the centre of a sphere and terminating at both ends in its surface, is called the Diameter; and a straight line drawn

from the centre to any point in the surface, is termed the Radius.

CASE I.

283. To find the convex surface of any prism or cylinder.

1. What is the convex surface of a cube whose side is 8 ft.?

OPERATION.

8 ft. x 4 32 ft. = perimeter of base.

=

32 ft. x 8 ft. = 256 sq. ft., Ans.

NOTE.-Any two opposite faces of a cube may be considered as its bases.

2. What is the convex surface of a cylinder whose altitude is 10 ft., and the diameter of its base 10 ft.?

OPERATION.

3.1416 x 10 ft. 31.416 ft. =

31.416 ft. x 10 ft. =

perimeter of base.
314.16 sq. ft., Ans.

RULE.-Multiply the perimeter of the base by the

altitude.

NOTE.-To find the entire surface, add the area of the bases to the convex surface.

3. Find the convex surface of a triangular prism whose altitude is 20 ft., and the sides of its base 8, 10, and 12 ft., respectively. Ans. 600 sq. ft.

4. How many sq. ft. of boards will it require to cover the entire outside of a cylindrical boiler, which is to be shipped to Boston, the diameter of the boiler being 3 ft. and the length 12 ft.? Ans. 213.39 sq. ft. 5. What is the entire surface of a cylindrical rollingstone, 5 ft. in length, and 15 in. in diameter ? Ans. 22.09 sq. ft.

CASE II.

284. To find the volume of any prism or cylinder.

1. What is the solidity of a triangular prism whose altitude is 15 ft., and the sides of its base, respectively, 3, 4, and 5 ft.?

OPERATION.

The area of the base is 6 sq. ft.

6 sq. ft. x 15 ft.

90 cu. ft. volume.

2. Find the volume of a cylinder whose altitude is 28 ft., and the diameter of its bases 2 ft.

21 ft. x .7854

OPERATION.

4.90875 sq. ft.

area of base.

4.90875 sq. ft. × 28 ft. = 137.445 cu. ft., Ans.

RULE.-Multiply the area of the base by the alti

tude.

3. What is the cost of a block of Carrara marble S ft. long, 3 ft. 10 in. wide, and 6 ft. 6 in. high, at $2.25 a cu. ft.? Ans. $448.50.

4. How many bushels of wheat can a cylindrical vessel hold, that is 4 ft. 6 in. in diameter and 5 ft. 4 in. deep? Ans. 68 bu. 5 qt.

CASE III.

285. To find the convex surface of a pyramid or

cone.

1. What is the convex surface of a square pyramid, whose slant height is 60 in., and each side of its base 30 in.?

30 in. x 4 = 120 in.

120 in. x (60 in.÷2)

OPERATION.

=

perimeter of the base.

3600 sq. in. 25 sq. ft., Ans.

2. Find the convex surface of a cone whose diameter is 8 ft., and the slant height 20 ft.

OPERATION.

8 ft. x 3.1416 25.1328 ft. circumference. 25.1328 ft. × (20 ft.÷2) =251.328 sq. ft., Ans.

RULE.-Multiply the perimeter of the base by onehalf the slant height.

NOTE.-Add the area of the base to the convex surface for the entire surface.

3. What is the entire surface of a pyramid whose base is a rectangle 12 ft. by 9 ft., and its slant height 8 ft. 6 in.? Ans. 2861 sq. ft.

4. The diameter of a cone is 20 in., and its slant height 20 ft.; what is its entire surface ?

CASE IV.

Ans. 54.54 sq. ft.

286. To find the solidity of a pyramid or cone.

1. What is the solidity of a square pyramid, the side of whose base is 9 ft., and the altitude 27 ft.?

OPERATION.

9 ft. x 9 ft. = 81 sq. ft. = area of base.

81 sq. ft. × (×27 ft.)

= 729 cu. ft., Ans.

2. What are the solid contents of a cone, the diameter of whose base is 4 ft., and the altitude 15 ft.?

OPERATION.

4 ft. x.7854 12.5664 sq. ft. area of base. 12.5664 sq. ft. x († × 15 ft.) = 62.832 cu. ft., Ans.

RULE.—Multiply the area of the base by one-third of the altitude.

3. Find the cost of a triangular pyramid of brown stone, whose altitude is 9 ft., each side of the base being 3 ft., at $2 per cu. ft. Ans. $29.23. 4. What is the solidity of a cone generated by the revolution of a triangle about its perpendicular, whose hypothenuse is 13 ft. and base 5 ft.?

CASE V.

Ans. 314.16 cu. ft.

287. To find the convex surface of a frustum of a pyramid or cone.

1. Find the convex surface of a frustum of a square pyramid whose slant height is 19 ft., each side of the upper base 7 ft., and of the lower base 12 ft.

OPERATION.

48 ft.+28 ft. = 76 ft. = sum of perimeters.
76 ft. x (× 19 ft.) = 722 sq. ft., Ans.

RULE.-Multiply the sum of the perimeters of the hases by one-half the slant height.

TE.-The entire surface is equal to the convex surface plus eas of the bases.