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EXAMPLES. 539. To find the convex surface of a prism or cylinder.

1. Find the area of the convex surface of a prism whose altitude is 7 ft., and its base a pentagon, each side of which is 4 ft.

OPERATION. 4 ft. x 5 = 20 ft., perimeter. 20 ft. x 7 = 140 sq. ft., convex surface.

2. Find the area of the convex surface of a triangular prism whose altitude is 8} ft., and the sides of its base 4, 5, and 6 ft., respectively.

OPERATION. 4 ft. + 5 ft. + 6 ft.

perimeter. 15 ft. x 81 =1272 sq.ft., convex surface.

3. Find the area of the convex surface of a cylinder whose altitude is 2 ft. 5 in., and the circumference of its base 4 ft. 9 in.

OPERATION. 2 ft. 5 in.=29 in. ; 4 ft. 9 in.=57 in.

57 x 29 = 1653 sq. in. 11 sq. ft. 69 sq. in., convex surface.

RULE. - I. To find the convex surface, multiply the perimeter of the base by the altitude.

II. To find the ENTIRE surface, add the area of the bases or ends.

4. If a gate 8 ft. high and 6 ft. wide revolves upon a point in its center, what is the entire surface of the cylinder described by it ?

5. Find the superficial contents, or entire surface, of a parallelopipedon 8 ft. 9 in. long, 4 ft. 8 in. wide, and 3 ft. 3 in. high.

6. What is the entire surface of a cylinder formed by the revolution about one of its sides of a rectangle that is 6 ft. 6 in. long and 4 ft. wide ?

Ans. 263.89 sq. ft. 7. Find the entire surface of a prism whose base is an equilatera? triangle, the perimeter being 18 ft.. and the altitude 15 ft.

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

1. Find the volume of a triangular prism whose altitude is 20 ft., and each side of the base 4 ft.

OPERATION.

The area of the base is 6.928 sq. ft.

6.928 sq. ft. X 20 = 138.56 cu. ft., volume. 2. Find the volume of a cylinder whose altitude is 8 ft. 6 in., and the diameter of its base 3 ft.

OPERATION.

32 x .7854 = 7.0686 sq. ft., area of base.
7.0686 sq. ft. X 8.5 = 60.083 cu. ft., volume.

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

3. What is the volume of a parallelopipedon whose base is 9.8 ft. by 7.5 ft., and its height 5 ft. 3 in.?

4. What is the volume of a log 18 ft. long and 11 ft. in diameter ? 5. Find the solid contents of a cube whose edges are 6 ft. 6 in.

PYRAMIDS, CONES, AND SPHERES.

541. A Pyramid is a body having for its base a polygon, and for its other faces three or more triangles, which terminate in a common point called the vertex.

Pyramids, like prisms, take their names from their bases, and are called triangular, pentagonal, etc. A regular pyramid has for its base a regular polygon and has its vertex in a perpendicular to the center of the base.

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Pyramid.

Frustum.

Cone.

Frustum,

542. A Cone is a body having a circular base, and whose convex surface tapers uniformly to the vertex.

A cone is a body conceived to be formed by the revolution of a right-angled triangle about one of its sides containing the right angle.

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

544. The Slant Height of a regular pyramid is the perpendicular distance from its vertex to one of the sides of the base ; of a cone, a straight line from the vertex to the circumference of the base.

545. The Frustum of a pyramid or cone is that part which remains after cutting off the top by a plane parallel to the base.

546. A Sphere is a body bounded by a uniformly curved surface, all the points of which are equally distant from a point within called the center.

547. The Diameter of a sphere is a straight line passing through the center of the sphere, and terminated at both ends by its surface.

548. The Radius of a sphere is a straight line drawn from the center to any point in the surface.

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EXAMPLES.

549. To find the convex surface of a regular pyramid or of a cone.

1. Find the convex surface of a triangular pyramid, the slant height being 16 ft., and each side of the base 5 ft.

OPERATION.

(5 ft. + 5 ft. + 5 ft.) x 16 ft. = 2= 120 sq. ft., convex surface.

PRAC, AR.—26

2. Find the convex surface of a cone whose diameter is 17 ft. 6 in., and the slant height 30 ft.

OPERATION.

17.5 ft. x 3.1416 = 54.978 ft., circumference;

54.978 ft. x 30 ÷ 2 = 824.67 sq. ft., convex surface.

RULE.-I. To find the convex surface multiply the perim eter or circumference of the base by one half the slant height. II. To find the entire surface, add to this product the area of the base.

3. Find the entire surface of a pyramid whose base is 8 ft. 6 in. square, and its slant height 21 ft.

4. Find the entire surface of a cone,the diameter of whose base is 6 ft. 9 in., and the slant height 45 ft. Ans. 512.9 sq. ft.

5. Find the cost of painting a church spire, at $.25 a sq. yd., whose base is a hexagon 5 ft. on each side, and the slant height 60 ft.

550. To find the volume of a pyramid or of a

cone.

1. What is the volume, or solid contents, of a square pyramid whose base is 6 ft. on each side, and its altitude 12 ft. ?

OPERATION.

6 ft. x 6 ft. x 12 ft. ÷ 3 = 144 cu. ft., volume.

2. Find the volume of a cone, the diameter of whose base is 5 ft., and its altitude 10 ft.

RULE. altitude.

OPERATION.

52 ft. x .7854 x 101⁄2 ÷ 3 = 68.72 cu. ft., volume.

Multiply the area of the base by one third the

3. Find the solid contents of a cone whose altitude is 24 ft., and the diameter of its base 30 inches.

4. What is the cost of a triangular pyramid of marble, whose altitude is 9 ft., each side of the base being 3 ft., at $2 per cubic foot? Ans. $29.23.

5. Find the volume and the entire surface of a pyramid whose base is a rectangle 80 ft. by 60 ft., and the edges which meet at the vertex are 130 ft. Ans. 192000 cu. ft., volume.

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

1. What is the convex surface of a frustum of a square pyramid whose slant height is 7 ft., each side of the greater base 4 ft., and of the less base 18 in. ?

OPERATION.
The perimeter of the greater base is 16 ft., of the less 6 ft.

16 ft. + 6 ft. x 7 = 2 = 77 sq. ft., convex surface. RULE. - I. To find the convex surface, multiply the sum of the perimeters, or circumferences, by one half the slant height.

II. To find the entire surface, add to this product the areas of both ends, or bases.

2. Find the convex surface of a frustum of a cone whose slant height is 15 ft., the circumference of the lower base 30 ft., and of the upper base 16 ft.

3. How many square yards are there in the convex surface of a frustum of a pyramid, whose bases are heptagons, each side of the lower base being 8 ft., and of the upper base 4 ft., and the slant height 55 ft. ?

Ans. 2563 sq. yd. 552. To find the volume of a frustum of a pyramid or cone.

1. Find the volume of a frustum of a square pyramid whose altitude is 10 ft., each side of the lower base 12 ft., and of the upper base 9 ft. OPERATION. -122+92=225; (225+ V144 x 81) x10-3=1110 cu. ft.

RULE. To the sum of the areas of both bases add the square root of the product, and multiply the result by one third of the altitude.

2. How many cubic feet are there in the frustum of a cone whose altitude is 6 ft., and the diameters of its bases 4 ft. ε nd 3 ft. ?

3. How many cubic feet are there in a piece of timber 30 ft. long, the greater end being 15 in. square, and the less 12 in. ?

4. How many cubic feet are there in the mast of a ship, its height being 50 ft., the circumference at one end 5 ft. and at the other 3 ft. ?

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