MATHEMATICS: Geometry 1. Determine the value of each interior angle of a regular dodecagon. 2. How many sides has a polygon if the sum of the interior angles is 2520° ? 3. How many sides has polygon if the sum of its interior angles equals the sum of its exterior angles? 4. How many diagonals are there in a hexagon? 5. The area of a circle circumscribing a regular octagon is 100π . Find the area of the octagon. 6. Find the area of a pentagon whose apothem is 10 cm. 7. Determine the area of a regular 6-point star polygon if the inner regular hexagon has 10 cm side. 8. One side of a parallelogram is 10 cm and its diagonals are 16 cm and 24 cm, respectively. Find its area. 9. A piece of wire is shaped to enclose a square whose area is 2169 in . It is then reshaped to form a rectangle

whose length is 15 in. Determine the area of the rectangle. 10. A rhombus has an area of 2320 cm . If one diagonal measures 20 cm, determine the length of each side. 11. Find the area of a circle inscribed in a rhombus whose perimeter is 100 cm and whose longer diagonal is

40 cm. 12. Find the area of an isosceles trapezoid if the measure of one angle is 135° and the lengths of the bases are

10 and 18. 13. Assuming that the earth is a sphere whose radius is 6400 km, find the distance along a 3-degree arc at the

equator of the earth’s surface. 14. In a circle whose radius is 12, find the area of a minor segment whose arc has a central angle of 60° . 15. A circle with radius 6 cm has half its area removed by cutting off a border of uniform width. Find the

width of the border. 16. What is the surface area of a cube whose space diagonal is 6. 17. A parallelepiped has an altitude of 20 inches. The base is a parallelogram with adjacent sides 8 inches and

12 inches and included angle 60° . Find its volume. 18. The base of a pyramid is a regular hexagon with each side equal to 6 m and an altitude of 10 m. What is

the volume of water it can hold when full and inverted? 19. The lateral area of the right circular water tank is 92 sq. cm. Its volume is 342 cu. m., determine its

radius. 20. Determine the volume of a right circular cone of radius 8 cm and slant height of 17 cm. 21. The frustum of a pyramid has a lower base 50 m by 10 m and an upper base of 40 m by 8 m. If the altitude

of the frustum is 6 m, determine its volume. 22. Determine the total surface of an icosahedron if the length of each edge is 5 inches. ECE 521: ECE Elective Page 1 of 5

MATHEMATICS: Geometry 23. Find the volume of a regular tetrahedron if the area of each face is equal to 173.2 2cm . 24. The volume of a sphere is 336 cmπ . Determine the surface area of the sphere. 25. A hemisphere whose radius is 12 inches is surmounted by a right circular cone with the same radius and

an altitude of 15 inches. Find the total surface area. 26. Two equal spheres overlap such that the surface of one sphere passes the centroid of the other sphere. If

the radius of each sphere is 2 m, find the volume common to both spheres. 27. Find the volume of a triangular spherical pyramid whose base angles are 95° , 105° and 135° on a sphere

of radius 30 cm. 28. A tank in the shape of a paraboloid is filled with water to height of 5 m and the diameter of the circular

section at the water surface is 3 m. Determine the volume of water inside the tank. 29. Each side of a cube is increased by 1%. By what percent is the volume of the cube increased. 30. Find the volume of the upper portion of a regular pyramid having a square base of 10 m by 10 m and an

altitude of 60 m starting at a point halfway of its altitude. 31. If the surface area of a sphere is increased by 30%, by what percent is the volume of the sphere increased? Exercises: Geometry 1. Three circles 1C , 2C and 3C are externally tangent to each other. Center to center distances are 10 cm

between 1C and 2C , 8 cm between 2C and 3C and 6 cm between 3C and 1C . Determine the total areas of the circles. (a) 2184.12 cm (b) 2157.08 cm (c) 2162.31 cm (d) 2175.93 cm

2. A rectangle is inscribed in a circle whose radius is 5 inches. The base of the rectangle is 8 inches. Find the area of the rectangle. (a) 44 (b) 50 (c) 48 (d) 40

3. The wall at one end of an attic takes the shape of trapezoid because of a slanted ceiling. The wall is 8 ft high at one end, 10 ft wide and only 3 ft high on the other end. Determine the area of the wall in sq. ft. (a) 50 (b) 55 (c) 60 (d) 48

4. The corner of a 2-meter square is cutoff to form a regular octagon. Determine the length of the resulting side of the octagon. (a) 0.834 (b) 0.724 (c) 0.732 (d) 0.828

5. A polygon has 170 diagonals. How many sides does it have? (a) 20 (b) 30 (c) 25 (d) 28

6. Find the length of the side of a pentagon if the line perpendicular to its side is 12 units from the center. (a) 14.74 units (b) 17.44 units (c) 71.44 units (d) 14.47 units

7. A regular dodecagon is inscribed in a circle of radius 24. Find the perimeter of the dodecagon. (a) 149.08 (b) 142.74 (c) 145.53 (d) 152.70

8. The length of the side of a square is increased by 100%. The perimeter is increased by ______. (a) 25% (b) 100% (c) 200% (d) 150%

9. The area of a rhombus is 168 2m . If one side of its diagonal is 12 m, find the length of the sides of a rhombus. (a) 15.23 cm (b) 10.42 cm (c) 14.22 cm (d) 12.43 cm

10. The perimeter of a sector is 9 and its radius is 3. What is the area of the sector? (a) 4.50 (b) 6.52 (c) 3.75 (d) 5.25

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MATHEMATICS: Geometry 11. If the altitude and base of a parallelogram are each increased by 5 inches. If the altitude is increased by 3

inches and the base is decreased by 2 inches, the area will increase by 5 sq. inches. Determine the original values in inches of the altitude and base of the parallelogram. (a) alt =3.2 in, base =0.8 in (c) alt =0.8 in, base =4.2 in (b) alt =1.0 in, base =3.0 in (d) alt =2.0 in, base =2.0 in

12. Three spheres of radii 10, 20 and 30 cm, respectively, are melted and formed into a single sphere. Find the volume of the single sphere. (a) 48,000π (b) 45,000π (c) 50,000π (d) 42,000π

13. The bases of a right prism are regular hexagons with each side equal to 6 cm. The bases are 10 cm apart. What is the volume of the prism? (a) 935.31 (b) 943.57 (c) 783.22 (d) 893.73

14. What is the measure of the interior angle of a regular 2000-gon? (a) 180° (b) 220° (c) 200° (d) 178°

15. Seven regular hexagons, each with 6-cm sides, are arranged so that they share some side and the centers of the six hexagons are equidistant from the seventh central hexagon. Determine the ratio of the area of one hexagon to the total outer perimeter enclosing the hexagons. (a) 0.8660 (b) 1.0392 (c) 0.6014 (d) 0.7217

16. The volume of a right prism with an altitude of 15 m and having an equilateral triangle as its base is equal to 234 3m . Determine the length of the side of the triangular base. (a) 6 (b) 7 (c) 8 (d) 5

17. The interior angles of a polygon are in arithmetic progression. The least angle is 120° and the common difference is 5° . Find the number of sides. (a) 8 (b) 9 (c) 11 (d) 10

18. If the sides of a parallelogram and an included angle are 6, 10 and 100° , respectively, find the length of the shorter diagonal. (a) 10.73 (b) 15.32 (c) 12.53 (d) 11.71

19. A sphere of radius 10 m and a right circular cone of base radius 10 m and height 15 m stands on a table. At what height from the table should the two solids be cut in order to have equal circular sections? (a) 3.14 m (b) 3.24 m (c) 3.44 m (d) 3.54 m

20. The sum of the interior angles of a polygon of n sides is 1080° . Find the value of n . (a) 10 (b) 9 (c) 8 (d) 11

21. A prism has an equilateral triangle with 20 cm on a side for its base, and an altitude of 30 cm. Determine the lateral area. (a) 1800 (b) 1750 (c) 1505 (d) 2100

22. Find the volume of a spherical wedge whose angle is 54° on a sphere of radius 27 cm. (a) 12,367.19 3cm (c) 12,422.42 3cm (b) 11,243.42 3cm (d) 10,626.71 3cm

23. A metal sphere is melted and recast into a hollow spherical shell whose outer radius is 277 cm. The radius of the hollow interior of the shell is equal to the radius of the original sphere. Find the radius of the original sphere. (a) 200 cm (b) 225 cm (c) 220 cm (d) 240 cm

24. Find the total surface area of a regular triangular pyramid if each edge of the base measures 6 inches and each lateral edge of the pyramid measures 5 inches. (a) 51.59 2in (b) 50.23 2in (c) 54.42 2in (d) 50.21 2in

25. The altitude of an oblique circular cone is 8 inches and its longest element is 17 inches. Find the length of the shortest element if its volume is 54π cubic inches. (a) 8 (b) 7 (c) 9 (d) 10

26. Find the area of the spherical lune whose angle is 75° on a sphere of radius 30 cm. (a) 2356.19 2cm (b) 2052.52 2cm (c) 2054.34 2cm (d) 2257.26 2cm

27. What is the name of a polygon that has 27 diagonals? (a) hexagon (b) decagon (c) nonagon (d) undecagon

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MATHEMATICS: Geometry 28. Water is poured to a depth of 12 cm into a hemispherical bowl of radius 20 cm. Find the volume of the

water. (a) 2210π (b) 2304π (c) 2104π (d) 2540π

29. The volume of a right circular cone is 36π . If its altitude is 3, find its radius. (a) 3 (b) 5 (c) 4 (d) 6

30. The volume of a regular pyramid whose base area is a regular hexagon is 156 3cm . If the altitude of the pyramid is 5 cm, determine the length of the sides of the base. (a) 4 cm (b) 5 cm (c) 3 cm (d) 6 cm

31. For a regular polygon of heptagon sides, find the number of degrees contained in each central angle. (a) 51.43° (b) 40° (c) 60° (d) 32.73°

32. The great pyramid of Egypt has a square base 232 m on a side and 147 m high. Find its volume. (a) 2,768,334 3m (b) 2,662,351 3m (c) 2,632,242 3m (d) 2,637,376 3m

33. A tetrahedron is a regular solid with a equilateral triangles for each of three surfaces. If each side is 10 cm, what is the volume of the tetrahedron? (a) 116.21 cu. cm (b) 117.85 cu. m (c) 91.67 cu. m (d) 83.33 cu. m

34. A canonical vessel has a height of 24 cm and a base diameter of 12 cm. It holds water to depth of 18 cm above its vertex. Find the volume of its content in 3cm . (a) 381.7 (b) 392.2 (c) 353.3 (d) 363.2

35. The area of a hexagon inscribed in a circle is 374.11 2cm . Determine the area of the circle. (a) 452.39 (b) 465.32 (c) 423.34 (d) 431.46

36. A polygon has 170 diagonals. How many sides does it have? (a) 16 (b) 22 (c) 18 (d) 20

37. If the edge of the cube is increased by 20%, find the percentage increase in volume? (a) 72.8% (b) 83.2% (c) 76.3% (d) 63.3%

38. The sum of the interior angles of a polygon is 540° . Find the number of sides. (a) 4 (b) 6 (c) 5 (d) 7

39. The height of a circular cone with circular base is h . If it contains water to a depth of 2/3h , what is the ratio of the volume of the water to that of the cone? (a) 1:27 (b) 8:27 (c) 26:27 (d) 24:27

40. The diagonal of the face of a cube is 3 2 cm. Find the main diagonal of the cube. (a) 3 3 (b) 2 3 (c) 2 2 (d) 4 2

41. Find the length of the diagonal of a cube whose volume is 729 cubic cm. (a) 15.59 cm (b) 17.43 cm (c) 12.54 cm (d) 14.72 cm

42. A room is 12 ft wide, 15 ft long and 8 ft high. If an air conditioner changes the air once every five minutes, how many cubic feet of air does it change per hour? (a) 17,280 (b) 19,553 (c) 14,522 (d) 16,733

43. The base areas of a frustum of a cone are 25 2cm and 16 2cm , respectively. If its altitude is 6 cm, find its volume. (a) 120 (b) 125 (c) 122 (d) 136

44. Find the length of the side of a regular pentagon inscribed in a circle of radius 10 cm. (a) 10.34 cm (b) 11.76 cm (c) 12.42 cm (d) 35.22 cm

45. The area of a zone of a spherical segment is 1/4 that of a sphere. What is the ratio of the radius to the altitude of the spherical zone? (a) 3:1 (b) 2:1 (c) 2:3 (d) 3:4

46. A cone and cylinder have the same height and the same volume. Find the ratio of the radius of the cone to the radius of the cylinder. (a) 2 (b) 2 (c) 3 (d) 3

47. If the edge of the cube is increased by 30%, by what percent is the surface area increased? (a) 45% (b) 90% (c) 70.7% (d) 69%

48. Find the area of a regular pentagon whose side is 25 cm and apothem is 17.2 cm. (a) 1075 (b) 1142 (c) 1010 (d) 1067

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MATHEMATICS: Geometry 49. The bases of a right prism are a hexagon with one side 6 cm long. If the volume of the prism is 450 3cm ,

how apart are the bases? (a) 4.81 cm (b) 4.22 cm (c) 5.32 cm (d) 5.06 cm

50. The circumference of the base of a right circular cylinder is 48 cm and its altitude is 15 cm. Determine its total surface area. (a) 1086.69 2cm (b) 1042.52 2cm (c) 1204.23 2cm (d) 1102.62 2cm

51. A trapezoid has an area of 36 2m and altitude of 2 m. Its two bases have ratio of 4:5. What are the lengths of the longer base? (a) 20 m (b) 30 m (c) 16 m (d) 25 m

52. The lateral area of a right circular cone is 4 times the area of the base. Find the angle at which an element of the cone is inclined to the base. (a) 76 45'° (b) 73 36'° (c) 75 31'° (d) 77 42'°

53. Three spheres of radii 10, 20 and 30 cm, respectively, are melted and formed into a single sphere. Find the surface area of the single sphere. (a) 12,645.37 2cm (c) 13,700.77 2cm (b) 10,432.38 2cm (d) 13,077.42 2cm

54. A sphere having a diameter of 30 cm is cut into 2 segments. The altitude of the first segment is 6 cm. What is the ratio of the area of the second segment to that of the first? (a) 2 (b) 4 (c) 3 (d) 2.5

55. What is the spherical excess of a spherical polygon of four sides whose angles are 95° , 112° , 134° and 78° ? (a) 60° (b) 54° (c) 67° (d) 59°

56. Two equilateral triangles, each with 12 cm sides overlap each other to form a 6-point star. Determine the overlapping area. (a) 60° (b) 54° (c) 67° (d) 59°

57. A reverse curve on a railroad track consists of two circular arcs. The central angle of one is 20° with radius 2500 ft and the central angle of the other is 25° with radius of 3000 ft. Find the total length of the two arcs. (a) 2182 ft (b) 2282 ft (c) 2382 ft (d) 2482 ft

58. The area of a spherical lune is 90 2cm . If the area of the sphere is 810 2cm , what is the angle subtended by the lune? (a) 45° (b) 40° (c) 48° (d) 42°

59. The diagonals of a rhombus are 10 cm and 8 cm, respectively. Find its area. (a) 20 (b) 60 (c) 40 (d) 80

60. Find the surface area of a right circular cone in which radius measures 14 inches while the slant height measures 20 inches. (a) 1520 2in (b) 1453 2in (c) 1496 2in (d) 1562 2in

61. Find the volume of a spherical cone in a sphere of radius 25 cm, if the radius of the zone is 10 cm. (a) 2735.8 (b) 2232.2 (c) 2563.43 (d) 2442.73

62. The angle of a sector is 30° and the radius is 15 cm. What is the area of the sector? (a) 58.9 2cm (b) 62.3 2cm (c) 45.6 2cm (d) 52.2 2cm

63. A regular octagon is inscribed in a circle of radius 10 inches. Find the area of the octagon. (a) 282.8 (b) 265.5 (c) 302.2 (d) 294.8

64. Find the volume of a hexagonal spherical pyramid whose base angles are 135° , 105° , 122° , 131° and 142° on a sphere of radius 20 in. (a) 1303.18 3in (b) 1243.22 3in (c) 1205.92 3in (d) 1043.32 3in

65. A hemisphere whose radius is 12 inches is surmounted by a right circular cone with the same radius and altitude of 15 inches. Find the total volume. (a) 5422.84 3in (b) 5032.62 3in (c) 4252.32 3in (d) 5881.06 3in

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