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10. Find the ratio of the area of an equilateral triangle inscribed in a circle to that of the inscribed regular hexagon.

11. The sides of a triangle are 42, 40, and 37 feet; find the sides of a second triangle similar to the first, but four times larger in area.

12. Prove that the area of the regular hexagon inscribed in a circle is of the area of the circumscribing hexagon.

13. Indicate upon the faces of a pyramid the trace of a plane which divides the lateral surface into two parts in the proportion of 2 to 3.

14. To divide the triangle ABC, by a line DE parallel to AC, into two parts proportional to the numbers 8 and 9.

15. To divide the triangle ABC into five equal parts by lines D11, DE, DE, &c. parallel to AC.

16. To divide the triangle ABC into five parts proportional to the numbers 5, 6, 7, 8, 11, by lines D1E1, D2E, &c. parallel to AC.

17. The total surface of a right circular cone is S, and its radius is R; find the length of the generating line.

18. The lateral surface of a cone is S, and the length of the generating line is G; find its volume.

19. If the slant height of a cone is equal to the radius of the base, prove that the lateral surface is equal to the surface of the base.

20. AB is the diameter of a semicircle whose centre is O; upon each of the radii, OA, OB, are described semicircles; find the volume generated by the surface between the three semicircles.

21. The interior radius of a circular tower is r, and the height is h; find the thickness of the wall when the volume of masonry is v.

22. The height of a cone is equal to its diameter; find the ratio of the surface of the base to the lateral surface.

23. The volume of the frustum of a cone = V, and the radii of the ends are R and r; find h.

24. From a sphere of radius r a segment is cut off, of which the convex surface is s; find the surface of its base.

25. Find the radius of a sphere whose volume is 2 cubic feet.

26. A cylinder circumscribes a sphere; find the ratios of the surface, and of the volume of the sphere to the total surface, and to the volume of the cylinder, r being the radius of the sphere.

27. An arc of a great circle of a° is ; find the volume of the sphere.

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28. If a the side of a regular pentadecagon inscribed in a circle whose radius is R, prove

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29. Shew how to find the volume generated by a triangle turning round any line in its plane.

30. The radii of two circles are r1 and r2, the distance of their centres is a; find the length of their common chord. Find what relation must exist between the constants that the chord may be zero.

31. Find the sides of a right-angled triangle, knowing that they are three consecutive numbers.

32. If a = the side of a regular polygon inscribed in a circle whose radius is R, and A = the side of the similar circumscribed polygon, prove

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33. In the fig. of Euclid, I. 1, if r denote the radius of each circle, find the area of the part common to both circles.

34 Three equal circles touch each other; find the area of the part between them, the radius of each circle being r.

35. Divide a circle of radius into three equal parts by concentric circles.

36. Shew that the area of the regular inscribed -octagon is equal to that of a rectangle whose adjacent sides are equal to the sides of the inscribed and circumscribed squares.

37. Find the ratio of the volumes of two cylinders whose lateral surfaces are equal.

: 38. Find the ratio of the lateral surfaces of two cylinders whose volumes are equal.

39. Shew that the areas of all triangles described about the same circle are proportional to their perimeters.

40. If α, ß, y denote the distances from the angular points of a triangle to the points of contact of the inscribed circle, shew that its radius is

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41. The perimeter of an equilateral triangle being 27 yards, find the area.

42. If the right angle were divided into sixty degrees instead of ninety, what would be the circular measure of 75°?

43. An arc of 120° in one circle is equal to the whole circumference of another; compare their radii, and find how many degrees of the circumference of the smaller circle are equal in length to the radius of the greater.

44. The diameter of a circle is divided in the ratio of 1: 2, and circles are described on the segments as diameters; compare the areas of the three circles.

45. Find the length of the arc which subtends an angle 6° 12′ 36′′ in a circle whose radius is 100 yards.

46. How many cubic inches of iron will be required to form a garden roller, which is inch thick, with

an outer circumference of 5 feet, and a length of 31 feet.

47. If r be the radius of the inscribed circle of a triangle whose sides are a, b, c, and Pa, Pu, P., the three perpendiculars, prove

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48. If a triangle, ABC, be formed by joining the centres of three circles which touch each other, and if s be half the sum of its sides, a, b, c, then the radii of the circles whose centres are A, B, C are respectively

s-a, s-b, s-c.

49. At the distance of 50 miles from a tower, its top just appeared in the horizon; find its height, having given the earth's diameter to be 7964 miles.

50. Find the area of a circle traced on a sheet of paper by a pair of compasses whose legs are 6 inches, and contain an angle of 120°; find also the area when the legs are at right angles to each other.

51. The hypothenuse, AB, of a right-angled triangle is trisected in D and E; prove that the sum of the squares of the sides of the triangle, CDE = AB2.

52. An equilateral triangle and a regular hexagon have the same perimeter; shew that the areas of their inscribed circles are as 4: 3.

53. If Par Po Po denote the lengths of the lines drawn from the angles of a triangle to the middle of the opposite sides, prove that side

c = 3√6(po2 + po2) — 3 p2.

54. In a given circle is inscribed an equilateral triangle, and in that triangle another circle, and so on ad infinitum; shew that the sum of the circumferences of all the inner circles is equal to that of the given circle, and the sum of their areas to a third of its area.

55. Shew that the sum of the squares of the distances

of the centre of the inscribed circle from the angular points of a triangle

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56. If r, T1, T2, 7s be the radii of the inscribed and escribed circles of a triangle, and R the radius of the circumscribing circle, prove

and that

Area of triangle = √rr1rrs,

r1+r2+rs-r=4R.

57. Shew that the chord of a quadrant of a circle divides the circle into two parts in the ratio of 10 : 1 very nearly.

58. For 1 inch of rainfall, shew that the weight of water on an acre of ground is 100 tons nearly.

59. A circular hole is to be cut in a circular plate, so that the weight of the plate is reduced; find the diameter of the hole.

60. The area of the regular hexagon inscribed in a circle is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.

61. If a spherical shell, when formed into a solid sphere, be equal in volume to its own cavity, shew that the thickness of the shell = R x .20629.

62. A cubic foot of copper is drawn into a wire th of an inch in diameter; find its length.

63. A spherical shell weighs 18ths of a solid sphere of the same size and material; find the diameter of the internal cavity.

64. A perfectly flexible rope of uniform thickness is closely coiled upon a level floor. If the diameter of the rope be 2a, shew that the length of the first n complete coils will be

Tan(2n+1);

shew also how to find the length of the (n + 1)th coil.

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