REVIEW – PLANE and SOLID GEOMETRY

1. In triangle ABC, a= 12 cm, c=14 cm, and b= 34.770. Find the area of the triangle.
2 2 2
a. 27.8 cm b. 37.5 c. 32.6 d. 47.92
2. A regular octagon is inscribed in a circle whose radius is 12. Find the area of the octagon.
a. 521.31 b. 407.29 c. 351.27 d. 351.25
3. Find the area of a regular hexagon whose sides measure 5 cm.
2 2 2 2
a. 64.95 cm b. 47.85 cm c. 96.71 cm d. 69.53 cm
4. The apothem of regular pentagon is 10. Determine its area.
a. 227.43 b. 159.62 c. 363.30 d. 315.23
5. Find the radius of a circle inscribed in a rhombus whose perimeter is 100 inches and whose longer diagonal is 40 inches.
a. 10 in. b. 12 in. c. 14 in. d, 18 in.
6. Find the area of a trapezoid whose median is 32 cm and whose altitude is 6 cm.
2 2 2 2
a. 150 cm b. 164 cm c. 142 cm d. 192 cm
7. Three circles with radii 3, 4 and 5 inches are tangent to each other externally. Find the area of the triangle formed by joining
their centers.
2 2 2 2
a. 198.45 in b. 107.33 in c. 110.45 in d. 101.64 in
8. Find the area of an equilateral triangle inscribed in a circle of radius 20 cm.
2
a. 519.61 cm2 b. 456.28 cm2 c. 621.46 cm2 d. 516.45 cm
9. If the lengths of the diagonal of a rhombus are 6 and 8, find the perimeter of the rhombus.
a. 14 b. 20 c. 30 d. 28
10. Two circles, each of radius 6 units have their centers 8 units apart. Find the length of their common chord.
√
a. 2 5 √
b. 3 5 c. 4 5 √ √
d. 5 5
11. What is the apothem of a regular polygon having an area 225 sq. cm and a perimeter 60 cm?
a. 7.5 cm b. 6.5 cm c. 8.5 cm d. 4.5 cm
12. Find the area of a regular hexagon of side 3 cm.
a. 22.28 cm2 b. 23.38 cm2 c. 24.48 cm2 d. 25.58 cm2
13. A triangle has sides 3, 6 and 9. Find the shortest side of a similar triangle whose longest side is 15.
a. 6 b. 10 c. 8 d. 5
14. The perimeter of an octagon is 32 and its longest side is 6. What is the longest side of a similar octagon whose perimeter is
24?
a. 3.5 b. 4 c. 4.5 d. 5
15. A hexagon is circumscribed about a circle of radius 5. If the perimeter of the hexagon is 38, what is the area of the hexagon?
a. 75 b. 65 c. 85 d. 95
16. Π Π
The circumferences of two circles are 6 and 10 . What is the ratio of their areas?
a. 9/25 b. 8/25 c. 7/25 d. 6/25
17. If a regular polygon has 54 diagonals, then it has how many sides?
a. 10 b. 11 c. 14 d. 12
18. If the perimeter of a rhombus is 40 and one of its diagonals is 12, find the other diagonal.
a. 16 b. 15 c. 18 d. 17
19. Find the area of the annulus bounded by the inscribed and circumscribed circles of an equilateral triangle with a side of length
6.
a. 11 Π b. 8 Π c. 10 Π d. 9 Π
20. Find the area of a regular octagon inscribed in a circle whose radius is 10 cm.
a. 822.8 cm2 b. 282.8 cm2 c. 828.2 cm2 d. 228.8 cm2
21. If the perimeter of a regular hexagon is 24, what is its apothem?
a. 3 3√ b. 4 3 √ c. 2 3 √ d. 5 3 √
22. Two sides of a parallelogram are 20 and 30 and the included angle is 36 degrees. Find the length of the longer diagonal.
a. 74.65 b. 64.75 c. 57.46 d. 47.65
23. The sides of a triangle are 17, 21 and 28. Find the length of the line segment bisecting the longest side and drawn from the
opposite angle.
Prepared by: Engr. Mark Paolo C. Abag, REE
 a. 11 b. 12 c. 13 d. 14
24. A circle is inscribed in an equilateral triangle. If the circumference of the circle is 3, find the perimeter of the equilateral triangle.
a. 9.246 b. 6.294 c. 2.946 d. 4.962
25. Two concentric circles each contain an inscribed square. The larger square is also circumscribed about the smaller circle. If
the circumference of the larger circle is 12 pi, what is the circumference of the smaller circle?
√
a. 6 2 pi √
b. 5 2 pi √
c. 4 2 pi √
d. 3 2 pi
26. The area of the sector of a circle having a central angle of 60 degrees is 24 Π . Find the perimeter of the sector.
a. 34.4 b. 35.5 c. 36.6 d. 37.7
27. What is the angle at center of a circle if the subtending chord is equal to two thirds of the radius?
a. 39.95deg b. 38.94deg c. 37.93deg d. 36.92deg
28. From a point outside of an equilateral triangle, the distances of the vertices are 12, 20 and 12 respectively. Find the length of
each side of the triangle.
a. 23.95 b. 22.85 c. 21.78 d. 20.68
29. Find the area of a regular five pointed star that is inscribed in a circle of the radius 10.
a. 121.62 b. 112.26 c. 122.16 d. 126.21
30. A regular five pointed star is inscribed in a circle of radius b cm. Find the area between the circle and the star.
a. 4.04 b2 b. 3.03 b2 c. 1.01 b2 d. 2.02 b2
31. Find the area of a regular 6 – pointed star of David inscribed in a circle of radius 5 m.
a. 35.4 m2 b. 43.3 m2 c. 34.6 m 2 d. 29.7m2
32. Given that the perimeter of a triangle is 100 cm. If the angles of the triangle are in the ration 3:5:7, find the longest side of the
triangle.
a. 28.53 cm b. 41.25 cm c. 38.04 cm d. 29.06 cm
33. The area of a triangle lot is 2,598.08 m2. If the sides of the lot are in continued proportion of 3:5:7, find the shortest side of the
lot.
a. 30 m b. 40 m c. 50 m d. 60 m
34. If each side of a cube is increased by 1%. By what percent is the volume of the cube increased?
a. 3.03% b. 5.01% C. 1.03% D. 5.04%
35. A sphere of radius r just fits into a cylindrical box. Find the empty space inside the box. (Ans. A)
3 3 3 3
a. 2 Π r /3 b. 8 Π r /9 c. 4 Π r /9 d. 20 Π r /27
36. Find the volume of a pyramid having a pentagonal base with sides each equal to 12 cm and an apothem of 8 cm. The altitude
of the pyramid is 36 cm. (Ans. C)
a. 2, 660 cm3 b. 2, 770 cm3 c. 2, 880 cm3 d. 2, 990 cm3
37. The lateral area of a right circular cone with a radius of 20 cm and a height of 30 cm is (Ans. A)
a. 2, 265.43 cm2 b. 2, 236.45 cm2 c. 2, 245.63 cm2 d. 2, 253.46 cm2
38. Find the volume of a regular square pyramid whose slant height is 10 and whose base edge is 12. (Ans. A)
a. 384 b. 374 c. 364 d. 354
39. The base of a prism is a rhombus whose sides are each 10 cm and whose shorter diagonal is 12 cm. If the altitude is 12 cm,
find its volume. (Ans. C)
a. 1, 132 cm3 b. 1, 142 cm3 c. 1, 152 cm3 d. 1, 162 cm3
40. Find the volume of a triangular prism whose altitude is 20 cm and whose sides are 6 cm, 8 cm and 12 cm. (Ans. A)
a. 426.61 cm3 b. 421.66 cm3 c. 461.26 cm3 d. 416.62 cm3
41. A stone is dropped into a circular tub 40 inches in diameter, causing the water therein to rise 20 inches. What is the volume of
the stone? (Ans. C)
3 3 3 3
a. 6, 000 Π in b. 7, 000 Π in c. 8, 000 Π in d. 9, 000 Π in
42. The base of a right parallelepiped is a rhombus whose sides are each 10 cm long and one of whose angles is 60 degrees. If
the altitude of the parallelepiped is 4 cm, find its volume. (Ans. B)
√
a. 100 3cm
3
√
b. 200 3 cm
3
c. 300 3cm√ 3
d. 400 3 cm √ 3

43. Find the lateral area of a pyramid whose altitude is 27 cm and whose base is a square 8 cm on a side. (Ans. B)
a. 437.62 cm2 b. 436.72 cm2 c. 432.76 cm2 d. 427.63 cm2
44. The diagonal of a cube is 2 √3 . Find its volume. (Ans. C)
a. 9 b. 7 c. 8 d. 6
Prepared by: Engr. Mark Paolo C. Abag, REE
45. Find the volume of the frustum of a regular triangular pyramid whose altitude is 3 and whose base edges are 4 and 8
respectively. (Ans. D)
√
a. 25 3 b. 26 3 √ c. 27 3 √ d. 28 3 √
46. Find the lateral area of the frustum of a regular square pyramid whose base edges are 6 and 12 and whose altitude is 4. (Ans.
D)
a. 150 b. 160 c. 170 d. 180
47. Find the lateral edge of a regular square pyramid whose slant height is 8 and whose base edge is 6. (Ans. C)
a. 6.54 b. 7.54 c. 8.54 d. 9.54
48. The base edges of a triangular pyramid are 12, 14 and 16. If its altitude is 22, what is the volume of the pyramid? (Ans. D)
a. 594.64 b. 564.94 c. 544.69 d. 596.44
49. The volume of the frustum of a right circular cone is 78 pi. The upper base radius is 2 and the lower base radius is 5. What is
the altitude of the frustum? (Ans. B)
a. 5 b. 6 c. 7 d. 8
50. The volume of a right circular cone having a slant height of 13 and altitude 12 is (Ans. A)
a. 100 Π b. 150 Π c. 200 Π d. 250 Π
51. Find the lateral area of a regular triangular pyramid whose base edge is 4 and its lateral edge is 6. (Ans. D)
√
a. 21 2 b. 22 2√ c. 23 2 √ d. 24 2 √
52. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and its altitude is 4. Find its lateral area.
(Ans. A)
a. 75 Π b. 85 Π c. 95 Π d. 65 Π
53. Find the volume of a sphere whose surface area is 64 Π . (Ans. A)
a. 256 Π /3 b. 254 Π /3 c. 252 Π /3 d. 250 Π /3
54. Find the lateral area of a right circular cone if its slant height is 22 and the circumference of its base is 8. (Ans. D)
a. 55 b. 66 c. 77 d. 88
55. What is the diameter of a sphere for which its volume is equal to its surface area? (Ans. B)
a. 5 b. 6 c. 7 d. 8
56. The lateral area of a regular pyramid is 2, 048 and the perimeter of the base is 128. Find the slant height. (Ans. C)
a. 42 b. 22 c. 32 d. 52
57. The area of the base of a right circular cone is 144 Π . If its altitude is 14, find its slant height. (Ans. A)
a. 18.44 b. 17.33 c. 16.22 d. 15.11
58. Find the volume of the largest circular cylinder that can be inscribed in a cube whose volume is 64 cu cm. (Ans. D)
3 3 3 3
a. 13 Π cm b. 14 Π cm c. 15 Π cm d. 16 Π cm
59. The lateral area of a regular pyramid is 514.5 and the slant height is 42. Find the perimeter of the base. (Ans. A)
a. 24.5 b. 26.5 c. 22.5 d. 28.5
60. Find the volume of a sphere that is circumscribed about a cube of edge 4. (Ans. B)
√
a. 30 3 Π b. 32 3 Π √ √
c. 34 3 Π d. 36 3 Π√
61. The lateral area of a regular pyramid is 2 048 and the perimeter of the base is 128. It is base is a regular octagon, find the
altitude of the pyramid (Ans. B)
a. 24.5 b. 25.5 c. 26.5 d. 27.5
62. Find the volume of the frustum of a pyramid whose bases are regular hexagons with base edges 5 cm and 10 cm respectively
and the altitude is 15 cm. (Ans. A)
a. 2, 273.31 cm3 b. 2, 171.33 cm3 c. 2, 327.13 cm3 d. 2, 713.32 cm3
63. What is the volume of a cube if the number of cubic units in its volume is twice the number of square units in its total surface
area? (Ans. D)
a. 1, 827 b. 1, 287 c. 1, 872 d. 1, 728
64. Find the lateral area of regular hexagonal pyramid whose lateral edges are each 13 cm and whose base has sides 10 cm
each. (Ans. B)
a. 350 b. 360 c. 370 d. 380
65. The ratio of the volumes of two spheres is 8:27. What is the ratio of their surface areas? (Ans. B)
a. 2/9 b. 4/9 c. 5/9 d. 7/9
66. If the diameter of a sphere is increased by 40%, by what percent is the volume increased? (Ans. D)
Prepared by: Engr. Mark Paolo C. Abag, REE
a. 144.7% b. 147.4% c. 177.4% d. 174.4%
67. The radii of two spheres are in the ratio 3:4 and the sum of their surfaces is 2, 500. Find the radius of the smaller sphere.
(Ans. B)
a. 14 b. 15 c. 16 d. 17
68. The volume of a rectangular parallelepiped is 162. The three dimensions are in the ratio 1:2:3. Find the total area. (Ans. A)
a. 198 b. 197 c. 196 d. 195
69. The base edge of a square pyramid is 3 m and its altitude is 10 m Find the area of the section parallel to the base and 6 m
from it. (Ans. C)
a. 1.22 m2 b. 1.33 m2 c. 1.44 m2 d. 1.55 m2
70. The area of the base of a pyramid is 25 and its altitude is 10. What is the distance from the base of a section parallel to the
base whose area is 9? (Ans. A)
a. 4 b. 3 c. 5 d. 2
71. A right circular cone whose slant height is 18 cm and the circumference of whose base is 6 cm is cut by a plane parallel to the
base such that the cone is cut off, has a slant height of 4cm. Find the lateral area of the frustum formed. (Ans. D)
a. 48.3 b. 49.1 c. 50.2 d. 51.3
72. A solid has a circular base of radius 20. Find the volume of the solid if every section perpendicular to a certain diameter is an
equilateral triangle. (Ans. A)
a. 18, 475.21 b. 14, 871.52 c. 17, 845.12 d. 15, 781.25
73. Find the total area of a regular hexagonal pyramid whose slant height is 5 ft and whose base is 4 ft. (Ans. C)
a. 105.71 ft2 b. 107.15 ft2 c. 101.57 ft2 d. 110.75 ft2
74. The slant height of the frustum of a right circular cone makes an angle of 60 degrees with the larger base. If the slant height is
30 cm and the radius of the smaller base is 5 cm find the volume of the frustum. (Ans. B)
a. 15, 283.7 cm3 b. 14, 283.7 cm3 c. 13, 283.7 cm3 d. 12, 283.7 cm3
75. The lateral area of the frustum of a regular pyramid is 336 sq. cm. If the lower base is a square having a side of 8 cm, the
upper base is a square of side x cm and its slant height is 12 cm, find the value of x. (Ans. A )
a. 6 b. 4 c. 7 d. 5
√ √
76. If the area of the base of a regular hexagonal prism is 3 3/2 sq. cm and the total area is 45 3 sq.cm, find the volume of
the prism. (Ans. B)
a. 20.5 cm3 b. 31.5 cm3 c. 21.5 cm3 d. 30.5 cm3
77. If a cylinder has a lateral area of 88 pi and a volume of 176 pi, what is its total area? (Ans. A )
a. 120 pi b. 125 pi c. 130 pi d. 135 pi
78. Find the volume of a spherical zone in a sphere of radius 17 cm if the radius of the zone is 8 cm. (Ans. D)
3 3 3 3
a. 1,126 Π /3 cm b. 1,136 Π /3 cm c. 1,146 Π /3 cm d. 1,156 Π /3 cm
79. The zone of a spherical cone has an altitude of 2 cm and a radius of 4 cm. Find the volume of the spherical cone. (Ans. C)
3 3 3 3
a. 115 Π /3 cm b. 110 Π /3 cm c. 105 Π /3 cm d. 1 00 Π /3 cm
80. Find the volume of a spherical segment if the radii of the bases are 3 and 4 respectively and its altitude is 2. (Ans. D)
a. 83.27 b. 87.32 c. 83.72 d. 82.73

Prepared by: Engr. Mark Paolo C. Abag, REE