RULE.

To find the convex surface of a frustum, multiply the sum of the perimeters or circumferences of the bases by one-half the slant height.

WRITTEN EXERCISES.

1. Find the convex surface of the frustum of a square pyramid whose slant height is 24 ft., and the side of the lower base 10 ft. and upper base 6 ft.

2. Find the convex surface of the frustum of a cone whose slant height is 42 ft., and the radii of the bases 12 and 4 ft., respectively.

3. Find the entire surface of the frustum of a cone whose slant height is 50 ft., and the radii of the bases 10 and 5 ft., respectively.

4. Find the entire surface of the frustum of a triangular pyramid whose slant height is 40 in., and the sides of the upper base 4 in. and of the lower base 10 in.

RULE.

263. To find the volume of a frustum, take the sum of the areas of the two bases, to which add the square root of their product, and multiply this sum by one-third of the altitude.

WRITTEN EXERCISES.

1. Find the volume of the frustum of a square pyramid the sides of whose bases are 3 ft. and 6 ft. and altitude 12 ft.

The volume

=

=

=

2. Find the contents of the frustum of a cone the radii of whose bases are 3 and 4 ft. and altitude 18 ft.

The contents

222 π

(32 + 62 + √32 × 62) 12 = (9 + 36 + 18) 4 ·

=

252 cu.ft.

=

Π

(π 32 + π 42 + Vπ2 32 × 42) 1⁄8 = (9 π + 16 π + 12 π) 6 697.4352 cu. ft.

3. Find the contents of the frustum of a square pyramid the sides of whose bases are 2 and 8 ft. and altitude 15 ft.

4. Find the contents of a log the radii of whose bases are 2 and 3 ft. and length 60 ft.

5. How many cubic feet in the frustum of a cone the radii of whose bases are 12 and 8 ft. and altitude 27 ft.?

6. Find the contents of the frustum of a triangular pyramid the sides of whose bases are 4 and 6 ft. and altitude 21 ft.

RULE.

The surface of a sphere is equal to the square of the diameter, or four times the square of the radius, multiplied by 3.1416.

WRITTEN EXERCISES.

1. Find the surface of a sphere whose radius is 5 in.

2. Find the surface of a sphere whose radius is 12 in.

3. Assuming the earth to be a sphere 7960 miles in diameter, how many square miles in its surface?

4. What will it cost to plate with gold a sphere 18 in. in diameter at $40.50 a square foot?

5. What will it cost to tin the hemispherical dome of an observatory 40 ft. in diameter, at 10 cents a square foot?

RULE.

265. The volume of a sphere is equal to one-sixth the cube of its diameter, multiplied by 3.1416.

WRITTEN EXERCISES.

1. Find the volume of a sphere whose radius is 10 in.

2. The outer diameter of a spherical shell is 12 in. and the inner diameter is 8 in.; find the contents.

3. Find the weight of a ball of gold 8 in. in diameter, if a cubic foot weighs 1204 lb.

4. Find the weight of a cannon-ball of cast iron 15 in. in diameter, if a cubic foot weighs 450 lb.

It is readily seen that

then

WRITTEN EXERCISES.

1. Find the side of the largest cube that can be cut from a sphere 15 in. in diameter.

A B2+ BC2 + CD2

3 A B

A

1

AD2. But A B = BC = CD;
A D2

15= 225,

A B2 = 75,

AB V75 - 8.66+ in.

2. Find the side of the largest cube that can be cut from a sphere 30 in. in diameter.

PROBLEMS IN MENSURATION.

1. How many acres in a triangle whose base is 40 chains and altitude 36 chains?

2. How many square yards in a triangle whose sides are

each 50 feet?

3. How many acres in a triangle whose sides are respectively 20, 30, and 40 chains?

4. The hypotenuse of a right triangle is 312 feet and the base is 282 feet; what is the perpendicular?

5. Find the diagonal of a square whose side is unity.

6. Find the diagonal of a cube whose side is unity.

7. How many acres in a parallelogram whose base is 161.5 rods and altitude 106.3 rods?

8. The area of a rectangle is 1872 square rods, and the length is to the width as 4 is to 3; required the length of the sides.

9. The area of a circle is 2 A. 3 R. 15 P.; what is the circumference?

10. Find the side of a square whose area is equal to the area of a circle 40 feet in diameter.

11. Find the length of an arc of 35° in a circle whose radius is 40 rods.

12. Find the area of the largest possible square that can be cut from a circle 62 feet in circumference.

13. Find the diameter of a circle whose area is numerically equal to 10 times its circumference.

14. How many square feet in the bottom and convex surface of a scrap-basket whose lower base is 9 inches in diameter, upper base 12 inches in diameter, and altitude 10 inches?

15. A piece of timber 90 feet long is 6 by 8 inches at the smaller end and 18 by 24 inches at the larger end; how many cubic feet does it contain?

16. Find the diameter of a sphere whose surface and volume are numerically equal.

17. Find the diagonal of a parallelopiped whose base is 5 feet by 7 feet, and altitude 16 feet.

18. Required the entire surface of a square pyramid whose altitude is 36 inches and side of the base 30 inches.

19. Required the dimensions of a cubical bin which will hold 1000 bushels of grain.

20. Find the contents of a cone whose altitude is 60 feet and the radius of the base 20 feet.

21. Find the volume of the frustum of a cone whose altitude is 24 feet, the radius of the lower base being 8 feet and of the upper base 6 feet.

22. If a conical hay-stack 24 feet in diameter and 20 feet high contains 12 tons, what are the dimensions of a similar stack which contains 20 tons?

23. A circular mirror 24 inches in diameter is surrounded by a frame 3 inches wide; what is the cost of the frame, at $2 a square foot?

24. A garden containing 74 square feet less than one-fourth of an acre is planted with hyacinths in squares 6 inches apart; how many plants will be required, if the outer rows are 6 inches from the edge?

25. The largest possible square piece of timber of uniform size is cut from a log 40 feet long, 15 inches in diameter at one end, and 18 inches at the other; how much is wasted?

26. A cylinder is 6 feet in diameter and 8 feet high; what are the dimensions of a similar cylinder whose surface is 64 times as much?

27. How much material will be required to make a hollow sphere whose outside diameter is 15 inches and thickness 1 inch?

28. What is the value of a gold sphere 4 inches in diameter, if a sphere of silver 1 inch in diameter is worth $5, and the value of gold is to the value of silver as 16 to 1?