8. If it cost $167.70 to enclose a circular pond containing 17 A. 110 P., what will it cost to enclose another as large?

9. If 63.39 rods of fence will enclose a circular field containing 2 acres, what length will enclose 8 acres in circular form?

REVIEW OF PLANE FIGURES.

PROBLEMS.

911. 1. How much less will the fencing of 20 acres cost in the square form than in the form of a rectangle whose breadth is the length, the price being $2.40 per rod?

2. A house that is 50 feet long and 40 feet wide has a square or pyramidal roof, whose height is 15 ft. Find the length of a rafter reaching from a corner of the building to the vertex of the roof.

3. Find the diameter of a circular island containing 14 sq. miles. 4. What is the value of a farm, at $75 an acre, its form being a quadrilateral, with two of its opposite sides parallel, one 40 ch. and the other 22 ch. long, and the perpendicular distance between them 25 chains?

5. Find the cost, at 18 cents a square foot, of paving a space in the form of a rhombus, the sides of which are 15 feet, and a perpendicular drawn from one oblique angle will meet the opposite side 9 feet from the adjacent angle.

6. A goat is fastened to the top of a post 4 ft. high by a rope 50 ft. long. Find the area of the greatest circle over which he can graze. 7. How much larger is a square circumscribing a circle 40 rods

in diameter, than a square inscribed in the same circle?

8. What is the value of a piece of land in the form of a triangle, whose sides are 40, 48, and 54 rods, respectively, at the rate of $125 an acre?

9. The radius of a circle is 5 feet; find the diameter of another circle containing 4 times the area of the first.

10. How many acres in a semi-circular farm, whose radius is 100 rods?

11. What must be the width of a walk extending around a garden 100 feet square, to occupy one-half the ground?

12. An irregular piece of land, containing 540 A. 36 P. is exchanged for a square piece of the same area; find the length of one of its sides? If divided into 42 equal squares, what is the length of the side of each?

13. A field containing 15 A. is 30 rd. wide, and is a plane inclining in the direction of its length, one end being 120 ft. higher than the other. Find how many acres of horizontal surface it contains.

14. If a pipe 3 inches in diameter discharges 12 hogsheads of water in a certain time, what must be the diameter of a pipe which will discharge 48 hogsheads in the same time?

SOLIDS.

912. A Solid or Body has three dimensions, length, breadth, and thickness.

The planes which bound it are called its faces, and their intersections, its edges.

913. A Prism is a solid whose ends are equal and parallel, similar polygons, and its sides parallelograms.

Prisms take their names from the form of their bases, as triangular, quadrangular, pentagonal, etc.

914. The Altitude of a prism is the perpendicular distance between its bases.

915. A Parallelopipedon is a prism bounded by six parallelograms, the opposite ones being parallel

916. A Cube is a parallelopipedon whose faces are all equal squares.

917. A Cylinder is a body bounded

Parallelopipedon.

Cube.

by a uniformly curved surface, its' ends being equal and parallel. circles.

1. A cylinder is conceived to be generated by the revolution of a rectangle about one of its sides as an axis.

2. The line joining the centers of the bases, or ends, of the cylinder is its altitude, or axis.

PROBLEMS.

918. 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 feet.

OPERATION.-4 ft. x 520 ft., peri

meter.

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 feet, and the sides of its base 4, 5, and 6 feet, respectively.

OPERATION. -4 ft. + 5 ft. + 6 ft. 15 ft., perimeter.

=

15 ft. x 81=127 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 in. x 291653 sq. in. = 11 sq. ft. 69 sq. inches, convex surface.

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

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?

7. Find the entire surface of a prism whose base is an equilateral triangle, the perimeter being 18 ft., and the altitude 15 ft.

919. 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 feet.

OPERATION.-The area of the base is 6.928 sq. ft. (882).

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 feet.

OPERATION.-32 × .7854 :

7.0686 square feet, area of base (905).

7.0686 sq. ft. x 8.5 60.083 cubic feet, volume.

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

3. Find the solid contents of a cube whose edges are 6 ft. 6 in. 4. Find the cost of a piece of timber 18 in. square and 40 ft. long, at $.30 a cubic foot.

5. Required the solid contents of a cylinder whose altitude is 15 ft. and its radius 1 ft. 3 in.

6. What is the value of a log 24 ft. long, of the average circumference of 7.9 ft., at $.45 a cubic foot?

PYRAMIDS AND CONES.

920. 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, square, or quadrangular, pentagonal, etc.

Pyramid.

Frustum.

Cone.

Frustum.

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

It is a body conceived to be formed by the revolution of a right-angled triangle about one of its sides containing the right angle, as an immovable axis.

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

923. The Slant Height of a pyramid is the perpendicular dis tance from its vertex to one of the sides of the base; of a cone, is a straight line from the vertex to the circumference of the base.

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

PROBLEMS.

925. To find the convex surface of a pyramid or

cone.

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

OPERATION. (5 ft.+5 ft. +5 ft.) × 16÷2 = 120 sq. ft., conv. surf. 2. Find the convex surface of a cone whose diameter is 17 ft. 6 in., and the slant height 30 feet.

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

To find the entire surface, add to this product the area of the base.

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

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

926. 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 feet on each side, and its altitude 12 feet.

OPERATION.-6 × 6 × 12÷3: =

144 cu. ft., volume.

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

OPERATION.-(52 ft. x .7854) × 10 ÷ 3 = 68.721 cu. ft., volume. RULE.-Multiply the area of the base by one-third the altitude. 3. Find the solid contents of a cone whose altitude is 24 ft., the diameter of its base 30 inches.

and

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 cu. foot?

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