|MCQs/Practice Test on Engineering Mathematics|
Engineering Mathematics - Geometry
1. Three circles C1, C2 and C3 are externally tangent to each other. Center to center distances are 10 cm between C1 and C2, 8 cm between C2 and C3 and 6 cm between C3 and C1. Determine the total areas of the circles.
(a) 184.12 cm^2
(b) 157.08 cm^2
(c) 162.31 cm^2
(d) 175.93 cm^2
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.
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.
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.
5. A polygon has 170 diagonals. How many sides does it have?
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.
8. The length of the side of a square is increased by 100%. The perimeter is increased by .
9. The area of a rhombus is 168 m^2. 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?
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
(b) alt = 1.0 in, base = 3.0 in
(c) alt = 0.8 in, base = 4.2 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.
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?
14. What is the measure of the interior angle of a regular 2000-gon?
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.
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 m^3. Determine the length of the side of the triangular base.
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.
18. If the sides of a parallelogram and an included angle are 6, 10 and 100°, respectively, find the length of
the shorter diagonal.
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 .
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.
22. Find the volume of a spherical wedge whose angle is 54° on a sphere of radius 27 cm.
(a) 12,367.19 cm^3
(b) 11,243.42 cm^3
(c) 12,422.42 cm^3
(d) 10,626.71 cm^3
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 in^2
(b) 50.23 in^2
(c) 54.42 in^2
(d) 50.21 in^2
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.
26. Find the area of the spherical lune whose angle is 75° on a sphere of radius 30 cm.
(a) 2356.19 cm^2
(b) 2052.52 cm^2
(c) 2054.34 cm^2
(d) 2257.26 cm^2
27. What is the name of a polygon that has 27 diagonals?
28. Water is poured to a depth of 12 cm into a hemispherical bowl of radius 20 cm. Find the volume of the water.
29. The volume of a right circular cone is 36π. If its altitude is 3, find its radius.
30. The volume of a regular pyramid whose base area is a regular hexagon is 156 cm^3. 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.
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 m^3
(b) 2,662,351 m^3
(c) 2,632,242 m^3
(d) 2,637,376 m^3
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 cm^3 .
35. The area of a hexagon inscribed in a circle is 374.11 cm^2. Determine the area of the circle.
36. A polygon has 170 diagonals. How many sides does it have?
37. If the edge of the cube is increased by 20%, find the percentage increase in volume?
38. The sum of the interior angles of a polygon is 540°. Find the number of sides.
39. The height of a circular cone with circular base is h. If it contains water to a depth of 2/3 h, what is the ratio of the volume of the water to that of the cone?
40. The diagonal of the face of a cube is 3sqrt(2) cm. Find the main diagonal of the cube.
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?
43. The base areas of a frustum of a cone are 25 cm^2 volume. and 16 cm^2, respectively. If its altitude is 6 cm, find its
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?
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.
47. If the edge of the cube is increased by 30%, by what percent is the surface area increased?
48. Find the area of a regular pentagon whose side is 25 cm and apothem is 17.2 cm.
49. The bases of a right prism are a hexagon with one side 6 cm long. If the volume of the prism is 450 cm^3, 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 cm^2
(b) 1042.52 cm^2
(c) 1204.23 cm^2
(d) 1102.62 cm^2
51. A trapezoid has an area of 36 m^2 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.
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 cm^2
(b) 10,432.38 cm^2
(c) 13,700.77 cm^2
(d) 13,077.42 cm^2
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?
55. What is the spherical excess of a spherical polygon of four sides whose angles are 95°, 112°, 134° and 78° ?
56. Two equilateral triangles, each with 12 cm sides overlap each other to form a 6-point star. Determine the overlapping area.
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 cm^2. If the area of the sphere is 810 cm^2, what is the angle subtended by the lune?
59. The diagonals of a rhombus are 10 cm and 8 cm, respectively. Find its area.
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 in^2
(b) 1453 in^2
(c) 1496 in^2
(d) 1562 in^2
61. Find the volume of a spherical cone in a sphere of radius 25 cm, if the radius of the zone is 10 cm.
62. The angle of a sector is 30° and the radius is 15 cm. What is the area of the sector?
(a) 58.9 cm^2
(b) 62.3 cm^2
(c) 45.6 cm^2
(d) 52.2 cm^2
63. A regular octagon is inscribed in a circle of radius 10 inches. Find the area of the octagon.
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 in^3
(b) 1243.22 in^3
(c) 1205.92 in^3
(d) 1043.32 in^3
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 in^3
(b) 5032.62 in^3
(c) 4252.32 in^3
(d) 5881.06 in^3