1. What is the area of a sphere whose diameter is 20 inches?

(20×3.1416)×20=1256.640=the surface.

EXAMPLES.

2. What is the surface of a sphere whose diameter is 30 inches?

3. What is the surface of a sphere whose circumference is 44 inches?

377. To find the solid content of a sphere,

RULE. Multiply the cube of the diameter by .5236. 4. What is the solidity of a sphere whose diameter is 16 inches?

163×.5236=2144.6656.

EXAMPLES.

5. What is the solidity of a sphere whose axis is 4 feet?

6. What is the solidity of the moon, supposing her to be a perfect sphere, and her axis 2180 miles?

378. To find the surface of a cylinder,

RULE. Multiply the circumference of the base by the height. The product will be the curve surface, to which add the areas of the ends.

7. What is the surface of a cylinder, whose length is 20 feet, and the diameter of the base 30 inches.

3.1416×2×20=157.080.

The area of the bases found by ART. 370.

8. What is the surface of a cylinder, whose length is 36 feet, and the circumference of its base 64 inches?

What is the rule for finding the solid content of a sphere? What is the rule for finding the surface of a cylinder?

379. To find the surface of a pyramid,

RULE. Find the area of the base, and the areas of the triangles which form the sides, separately. Their sum will be the whole surface.

9. What is the surface of a triangular pyramid, each side of the base of which is 30 inches, and the perpendicular upon a side from the vertex 12 feet ?

(6×21)X3=45 ft.-the surface of the sides.

The area of the base found by ART. 363.

10. What is the surface of a square pyramid, each side of the base of which is 40 inches, and the perpendicular upon a side from the vertex 16 feet?

380. To find the solidity of a pyramid, or cone,RULE. Multiply the area of its base by one third of its height or length.

EXAMPLES.

11. What is the solid content of a pyramid, whose height is 60 feet, and whose base is 20 feet square?

12. What is the solid content of a cone, whose height is 200 feet, and whose base is 50 feet in circumference?

381. To find the solidity of a prism, or cylinder, -

RULE. Multiply the area of the base by its length or height.

13. How many cubic feet in a cylinder, whose length is 10 feet, and the area of its base 4 feet?

10×4-40 feet.

What is the rule for finding the surface of a pyramid? What is the rule for finding the solidity of a pyramid, or cone? What is the rule for finding the solidity of a prism, or cylinder ?

EXAMPLES.

14. How many cubic feet in a stick of timber, 9 inches by 10 inches, and 50 feet in length?

15. How many cubic feet in a stick of timber, 10 inches by 14 inches, and 40 feet in length?

382. To find the surface of a frustum of a pyramid, or a cone,

RULE. Multiply half the sum of the circumference of both ends by the slant height, which will give the area of the curve, or convex, surface. The areas of both ends, added to this, will give the whole surface.

16. What is the surface of a frustum of a square pyramid, the sides of the bases being 40 and 30 inches, and the slant height 12 feet?

(40 + 30) × 2 × 12 1680 in. surface of slant height. (40 x 40) 12 133.3 in. of one ÷ = base. (30 x 30) 1275 in. of the other base. (1680133.3 + 75) ÷ 12 = 157.358 square ft.,

whole surface.

EXAMPLES.

17. What is the whole surface of a frustum of a pyramid of five equal sides, the perpendicular height and the sides of the base at one end 36 inches, at the other 24 inches?

18. What is the whole surface of a frustum of a cone, the diameter of the larger base being 20 inches, of the smaller base 8 inches, and the slant height 12 feet?

383. To find the solidity of a frustum of a pyramid, or a cone,

RULE. Find the area of the two ends, and add the

What is the rule for finding the surface of a frustum of a pyramid, or a cone? What is the rule for finding the solid contents of a frustrum of a pyramid, or a cone?

square root of their product to the sum of the areas of both ends. This, multiplied by one third of the perpendicular height, will give the solid content.

EXAMPLES.

19. What is the solid content of the frustum of a square pyramid, the sides of the bases being 15 and 6 feet, and the height 24 feet?

20. What is the solid content of the frustum of a triangular pyramid, the height of the frustum 14 feet, the sides of the greater base 21, 15, and 12, and those of the less base 14, 10, and 8 feet?

21. What is the solid content of the frustum of a cone, the diameters of the bases being 38 and 27 inches, and the height 11 feet?

22. What is the solid content of a mast of a ship, 60 feet high, and girths at one end 64 inches, and at the other 36 inches?

SIMILAR SURFACES.

SECTION XLII.

384. THE áreas of all circles are to each other as the square of their diameters, or as the square of their circumferences. Thus, if the diameter of one circle be 3 feet, and the diameter of another be 4, the first circle is to the second as 32: 42, or as 9: 16; the first circle is of the second.

16

If the circumference of one circle be 6 feet, and that of another 10 feet, the first circle is to the second as 62 102, or as 36: 100; the first circle is of the second.

100 25

What is the relation of different areas of circles?

EXAMPLES.

1. The diameter of a circle is 4 feet. What is the diameter of another, 6 times as large?

2. The diameter of a circular pond is 40 feet. What is the diameter of another, 5 times as large?

3. The circumference of a circle is 16 feet. What is the circumference of a circle 8 times as large?

4. The circumference of a pond is 150 feet. What is the circumference of a pond 6 times as large?

5. The circumference of a circular piece of land is 35 rods. What is the circumference of a piece 21 times as large?

385. The areas of all similar surfaces are to each other as the squares of their like dimensions.

386. All solid bodies are to each other as the cube of their diameters or similar sides.

6. If a ball 4 inches in diameter weigh 40 pounds, what will a ball of the same metal weigh, whose diameter is 6 inches?

7. If a ball 6 inches in diameter weigh 135 pounds, what will a ball weigh whose diameter is 4 inches? 8. What must be the inside measure of a cubical bin containing 20 bushels? (268.8 cu. in. to the gall.) 9. What must be the inside measure of a cubical cistern, that will hold 50 hogsheads?

10. If a ball 10 inches in diameter weigh 240 pounds, what is the diameter of one of the same metal that weighs 45 pounds?

What is the relation of the different areas of all similar surfaces ? What is the proportion of all solids to each other?