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Hence the surfaces of spheres are to each other as the squares of their radii.

Def.-A lune is that portion of a sphere contained between two semi-circumfer

ences of great circles, and terminated by the same diameter; the solid angle formed by the planes of the semicircles is said to be the angle of the lune.

It is shewn that two lunes are to each other as their angles; hence:

A

B

P

PROP. X.-The ratio of the surface of the lune to the whole surface of the sphere is the same as the ratio of the angle of the lune to four right angles; or, in a sphere whose radius is r, if A denote the ratio of the angle of the lune to a right angle, we have

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PROP. XI.-The surface of a spherical triangle is to that of the sphere as the sum of the angles of the triangle, diminished by two right angles, is to eight right angles.

Let ABC be the spherical triangle. Produce the sides AC and BC, till they intersect the circumference of which the third

side AB is a part. We then

have

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E

B

D

Adding, we have

2ABC + ABC + BCD + CDE + ACE = lune A + lune B

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.. ABC = (lune A + lune B+ lune C) – surface of

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+

+

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2 surface of sphere surface of sphere

Denoting the surface of the sphere by S, we have
ABC lune A lune B lune C

=

S

+

2S

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or as in the last proposition,

ABC
S

A+B+C - 2

8

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Exercises (7).

1. Find the total surface of a parallelopipedon which

measures

(1.) 10 inches, 15 inches, 18 inches.

(2.) 2 feet 3 inches, 5 feet 4 inches, 6 feet 10 inches. (3.) 6 feet 1 inch, 2 feet 10 inches, 1 foot 8 inches. (4.) 9 feet 5 inches, 3 feet 5 inches, 2 feet 4 inches. (5.) 6 feet 3 inches, 5 feet 4 inches, 4 feet 9 inches. (6.) 10 feet 1 inch, 1 foot 9 inches, 4 feet 3 inches.

2. Find the surface of a cube whose edge is

(1.) 15 inches.

(2.) 3 feet 4 inches.

(3.) 5 feet 10 inches.

(4.) 6 feet 3 inches.

(5.) 5 feet 1 inch.

(6.) 10 feet 3 inches.

feet.

3. Find the surface of a cube whose diagonal is 5 4. Find the edge of a cube which has a total surface

of 216 square inches.

5. Find the total surface of a triangular prism, the sides of whose base and height are

(1.) 3 inches, 4 inches, 5 inches, and 4 feet. (2.) 2 inches, 3 inches, 3 inches, and I foot. (3.) 5 feet, 5 feet, 5 feet, and foot.

(4.) 2 feet 3 inches, 1 foot 5 inches, 1 foot 5 inches, and foot.

(5.) 2 feet 6 inches, 3 feet 6 inches, and 15 inches. (6.) I foot 9 inches, i foot 9 inches, 1 foot 9 inches, and I foot 9 inches.

6. What must be the height of a triangular prism, each edge of the base being 23 inches, so that the total surface is I square foot?

7. A triangular prism of 10 inches in height stands. on an equilateral base; what must be the edge of the base when the total surface is 3 square feet?

8. Find the lateral surface of a hexagonal prism, each edge of base being 2 feet 3 inches, and height 5 feet.

9. Find the total surface of a cylinder whose radius. and height are

(1.) 2 inches, and 1 foot 4 inches.

(2.) 2 feet 3 inches, and 2 feet 3 inches.
(3.) 1 foot 6 inches, and 9 inches.
(4.) 9 inches, and 1 foot 6 inches.
(5.) I foot 3 inches, and 3 feet 9 inches.
(6.) 10 feet 6 inches, and 2 feet 4 inches.

10. Find the height of a cylinder whose lateral surface is equal to the surface of the two ends.

II. Find the height, in inches, of a cylinder standing on a base 2 inches in radius, and whose total surface is I square foot.

12. A cylinder 8 inches in height has a total surface of 1 square feet; find the radius of the base.

13. The lateral surface of a cylinder is 33 square feet, and the height is 6 inches; find the surface of the ends.

14. The total surface of a cylinder is 20 square feet,

and the height is equal to the diameter of the base; find the area of the base.

15. A cylinder whose height is equal to the radius of the base, has a total surface equal to that of a cube whose edge measures 9 inches; find the height of the cylinder.

16. Find the surface of a 4-inch drain pipe, 1 yard in length.

17. The total surface of a cylinder is 10 square feet, and the height is half the radius; find the height and the area of the two ends.

18. Find the total pressure on a cylindrical vessel measuring 5 feet in diameter, and 15 feet in length, there being 100 lbs. to the square inch.

19. Find the total surface of a square pyramid, the edge of whose base measures 3 inches, and the perpendicular height of the vertex is 5 inches.

20. Find the total surface of a square pyramid, the edge of whose base measures 3 inches, and the slant edge 5 inches.

21. Find the total surface of a square pyramid whose edge measures 3 feet 6 inches, and slant height 5 feet 4 inches.

22. Find the lateral surface of a cylinder whose diameter is 8 feet 3 inches, and length 6 feet 8 inches.

23. Find the total surface of the frustum of a pyramid whose ends are squares measuring 5 and 3 inches in the side, and the slant height is 12 inches.

24. Find the lateral surface of a cone whose diameter is 13 feet 6 inches, and slant side 18 feet 8 inches.

25. Find the lateral surface of a cone whose diameter is 17 feet 2 inches, and altitude 21 feet.

26. Find the lateral surface of a cone whose altitude is 1 yard, and diameter yard, and the cost of gilding the same at d. per square inch, including the base.

27. The sides of a triangle are 3, 4, 5 inches; if it revolve round the hypothenuse, find the whole surface generated by the other two sides.

28. Find the surfaces generated if the same triangle revolves round each of the other sides.

29. If a, b, c be the sides of a right-angled triangle, shew that the area generated by the revolution of the triangle round the hypothenuse is equal to the sum of the areas generated by revolving round the sides.

30. Find the lateral surface of the frustum of a cone whose top and bottom diameters are 5 feet 4 inches and 8 feet 3 inches, the slant height being 10 feet 4 inches.

31. Find the lateral surface of the frustum of a cone whose top and bottom diameters are 15 feet 4 inches, and 26 feet 8 inches, and the depth 23 feet 9 inches.

32. Find the lateral surface of a bucket whose top diameter measures 2 feet 4 inches, and the bottom diameter I foot 6 inches, the depth being 1 foot 9 inches.

33. Compare the lateral surfaces of a cone and cylinder standing upon the same base, and having the same vertical height.

34. The lateral surface of a right cone is 650 square inches, and the slant height is 25 inches; find the diameter of the base.

35. The lateral surface of a right cone is 10 square feet, and the radius of the base is 1 foot 4 inches; find the slant height.

36. The lateral surface of a right circular cone is 750 square inches, and the circumference of base is 50 inches; find the vertical height of the cone.

37. How many yards of canvas ths of a yard wide will be required for a conical tent 20 feet in diameter, and 15 feet high?

38. Divide a right cone into two parts by a plane parallel to the base, so that the lateral surfaces of the parts are equal.

39. Find the surface of a sphere of which the radius is

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(3.) 3 feet 4 inches.

(4.) 2.6 feet.

(5.) yard.

(6.) 3 feet.

40. What is the radius of a sphere of which the surface is

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