SOLIDS

233. A solid or body has length, breadth, and thickness. A prism is a solid whose sides are parallelograms, and whose two ends are equal polygons parallel to each other. Prisms, according to the form of their bases, are triangular, quadrangular, pentagonal, etc.

TRIANGULAR
PRISM

QUADRANGULAR PRISM

CUBE

A triangular prism is one that has triangles for its bases. A quadrangular prism is one that has quadrilaterals for its bases. Quadrangular prisms are also called parallelopipeds.

A cube is a solid with six equal square faces.

CYLINDER

A cylinder is a solid with a curved surface and two equal parallel circular bases.

The convex surface of a cylinder is the area of its curved surface.

The axis of a cylinder is the line joining the center of the two bases.

The altitude of a prism or a cylinder is the perpendicular distance between its bases.

A pyramid is a solid whose base is a polygon, and whose sides are triangles meeting at the vertex.

The altitude of a pyramid is the distance from the vertex to the center of the base; as AB in the figure.

The slant height of a pyramid is the distance from the vertex to the middle of the base of any of its triangular sides; as AC in the figure.

A cone may be regarded as a pyramid whose base is a circle. A sphere is a solid with a curved surface, every point of which is equally distant from a point within called the center.

The diameter of a sphere is a straight line passing through the center of the sphere and terminating in its surface. One half the diameter is the radius of the sphere. The greatest distance around the sphere is its circumference.

All the surface of a solid except its base or bases is called the lateral surface. The entire surface is the sum of the lateral surface and of the bases.

NOTE TO THE TEACHER: Derive the rules for finding the area of the surface of different solids in manner similar to the one that follows:

Cover exactly the lateral surface of any cylinder with a sheet of paper of the same height. Unroll the paper. Find its area. What is the circumference of the cylinder? Multiply the circumference of the cylinder by its

C

B

height. How does the product compare with the area of the paper? State the rule for finding the lateral surface of a cylinder.

B

Α'

Β'

The cylinder in the illustration is represented as lying on a plane touching it along the line AB. Conceive it rolled to the right. When A reaches the plane at A', B is exactly at the point B', and the cylinder has made one rotation. The path de

scribed by it is a rectangle equal to the entire lateral surface of the cylinder.

ORAL

234. 1. What is the lateral surface of a cylinder 8 inches in circumference, 14 inches high? Of a cylinder 10 inches in circumference, 10 inches high?

To find the surface of solids.

RULES: I. The lateral surface of a prism or a cylinder equals the product of the perimeter or circumference of the base by the altitude. II. The lateral surface of a pyramid or a cone equals one half the product of the perimeter of the base by the slant height.

III. The convex surface of a sphere equals the product of the diameter by the circumference, or it equals the area of four times the circle having the diameter of the sphere.

WRITTEN

2. What is the lateral surface of a triangular prism whose sides are 3 in., 4 in., and 5 in., and whose slant is 20 in.? What is the entire surface of this prism?

3. What is the lateral surface of a square prism whose base is 4 in. and altitude 9 in.? What is the entire surface?

4. What is the lateral surface of a cylinder with an altitude of 16 in. and a diameter of 12 in.? What is the entire surface?

5. What is the lateral surface of a cylinder having an altitude of 20 in. and a radius of 8 in.? What is the entire surface ?

6. What is the lateral surface of a square pyramid with a 12-in. base and 20-in. slant height? The entire surface?

7. What is the lateral surface of a pyramid whose base is an equilateral triangle having 12-in. sides, and whose slant height is 18 in.?

8. What is the lateral surface of a cone 20 in. in diameter with a slant height of 40 in.? The total surface?

9. What is the total surface of a conical tower 40 ft. high, whose radius is 30 ft.?

10. What is the surface of a 10-in. sphere?

11. What is the difference between the surfaces of a 20in. sphere, a 20-in. cube, and a circle 20 in. in diameter?

REVIEW
WRITTEN

235. 1. The length of the base of a parallelogram is 60 ft., its altitude is 39 ft. Find its area.

2. How many acres are there in a farm in the form of a parallelogram 45 rods wide and 68 rods long?

3. What is the area of a trapezoid whose parallel sides are 42 rods and 68 rods, the perpendicular distance between them being 371⁄2 rods?

4. The area of a trapezoidal field contains 6 acres; the bases are 300 ft. and 140 ft. respectively. Find the perpendicular distance between the sides.

5. The area of a trapezoid is 32,600 sq. in.; the perpendicular distance between the two bases is 76 in. If the length of one of the parallel sides is 120 in., what will be the length of the other parallel side?

6. What is the area of a triangle whose base is 67 ft. and altitude 19 ft. ?

7. What is the area of a triangular farm with a base of 62 rods and a perpendicular distance of 95 rods from the vertex to the base?

8. What is the cost of a triangular piece of land with a base of 2112 ft. and an altitude of 1650 ft., at $50 per acre? 9. A triangular farm containing 50 acres has a base of 60 rods.

Find its altitude.

10. What is the area of a circle whose diameter is 8 in. ? Of a circle whose radius is 7 ft. 6 in.? Find the area of

a circle whose circumference is 62.832 ft.

11. How much cloth is wasted in cutting the largest circle possible from a piece of cloth 16 in. square?

12. What is the difference in size between an 8-inch circle and a 16-inch circle?

13. Find the total atmospheric pressure, at 15 pounds per square inch, upon the top of a circular glass plate 32 in. in diameter.

14. Two circles, 5 in. and 12 in. respectively in diameter, have the same center. What is the area of the ring between their circumferences?

15. The area of a circle is one square inch. What is its diameter?

16. How long a rope will it take to fasten a horse to a post so that he may graze over one half acre of grass and no more, providing he can graze 2 ft. beyond the end of the rope?

17. Find the entire surface of a 6-inch cube; of a 12-inch cube. Find the diagonal of each of these cubes.