Name: _____________________________________________
Practice Problems (Plane and Solid Geometry)
1. The sum of the interior angles of a dodecagon is:
A. 2160°
B. 1980°
C. 1800°
D. 2520°
2. The diagonals of a parallelogram are 18 cm and 30 cm
respectively. One side of a parallelogram is 12 cm. Find the area of
the parallelogram.
A. 214
B. 216
C. 361
D. 108
3. How many diagonals can be drawn from a 12 sided polygon?
A. 66
B. 48
C. 54
D. 36
4. Find the area of a regular polygon whose side is 25m and
apothem is 17.2m.
A. 1075
B. 925
C. 1175
D. 1275
5. Find the area of a pentagon which is circumscribing a circle having
an area of 420.60 sq. cm.
A. 386.57
B. 450.54
C. 486.29
D. 260.24
6. In a triangle ABC, angle B has a size of 50o, angle A has a size of
32o and the length of side BC is 150 units. The length of side AB is
A. 232
B. 280
C. 260
D. 270
7. Find the area of a hexagon with a square having an area of 72 sq.
cm. inscribed in a circle which is inscribed in a hexagon.
A. 124.71 sq. cm.
B. 150.26 sq. cm.
C. 150.35 sq. cm.
D. 130.77 sq. cm.
8. The length of the side of a square is increased by 100%. Its
perimeter is increased by:
A. 25%
B. 100%
C. 200%
D. 300%
9. One side of a parallelogram is 10 m and its diagonals are 16 m and
24 m, respectively. Its area is:
A. 156.8 sq. m.
B. 185.6 sq. m.
C. 158.7 sq. m.
D. 142.3 sq. m.
10. Two squares each of 12 cm sides overlap each other such that
the overlapping region is a regular polygon. Determine the area of
the overlapping region thus formed.
A. 110.9 cm2
B. 119.3 cm2
C. 121.5 cm2
D. 117.4 cm2
11. If three sides of a triangle of an acute triangle is 3 cm, 4 cm, and
“x” cm, what are the possible values of x?
A. 1 < x < 5
B. 0 < x > 5
C. 0 < x < 7
D. 1 < x > 7
12. Two sides of a triangle measure 8 cm and 12 cm. find its area if
its perimeter is 26 cm.
A. 21.33 sq. m.
B. 32.56 sq. cm.
C. 3.306 sq. in.
D. 32.56 sq. in.
13. A regular hexagon with an area of 93.53 square centimeters is
inscribed in a circle. The area in the circle not covered by hexagon is:
A. 18.38 cm2
B. 16.72 cm2
C. 19.57 cm2
D. 15.68 cm2
14. The area of a circle is 89.4 square inches. What is the
circumference?
A. 35.33 inches
B. 32.25 inches
C. 33.52 inches
D. 35.55 inches
15. The corresponding sides of two similar triangles are in the ratio
3:2. What is the ratio of their areas?
A. 3
B. 2
C. 9/4
D. 3/2
16. The area of a rhombus is 132 square cm. if its shorter diagonal is
12 cm, the length of the longer diagonal is:
A. 20 centimeter
B. 21 centimeter
C. 22 centimeter
D. 23 centimeter
17. The length of the side of a rhombus is 5 cm. If the shorter
diagonal is of length 6 cm. What is the area of the rhombus?
A. 24 cm2
B. 14 cm2
C. 18 cm2
D. 25 cm2
18. If the radius of the circle is decreased by 20%, by how much is its
area decreased?
A. 36%
B. 26%
C. 46%
D. 56%
19. A sector has a radius of 12 cm. if the length of its arc is 12 cm, its
area is:
A. 66 sq. cm.
B. 82 sq. cm.
C. 144 sq. cm.
D. 72 sq. cm
20. A circle is divided into two parts by a chord, 3 cm away from the
center. Find the area of the smaller part, in cm square, if the circles
has an area of 201 cm square.
A. 51.4
B. 57.8
C. 55.2
D. 53.7
21. Two triangles have equal bases. The altitude of one triangle is 3
units more than its base and the altitude of the other is 3 units less
than its base. Find the altitudes, if the areas of the triangle differ by
21 square units.
A. 5 & 11
B. 4 & 10
C. 6 & 12
D. 3 & 9
22. The corners of a 2-meter square are cut off to form a regular
octagon. What is the length of the sides of the resulting octagon?
A. 0.525
B. 0.626
C. 0.727
D. 0.828
23. Two circles with radii 8 and 3 m are tangent to each other
externally. What is the distance between the points of tangency of
one of their common external tangencies?
A. 7.8 m
B. 9.8 m
C. 10.7 m
D. 6.7 m
24. A triangular piece of wood having a dimension 130 cm, 180 cm,
and 190 cm is to be divided by a line bisecting the longest side
drawn from its opposite vertex. The area of the part adjacent to the
180-cm side is:
A. 5126 sq. cm.
B. 5162 sq. cm.
C. 5612 sq. cm.
D. 5216 sq. cm.
25. Each interior angle of a regular polygon is 165°. How many sides?
A. 23
B. 24
C. 25
D. 26
26. A regular triangular pyramid has an altitude of 9 m and a volume
of 187.06 cu.m. What is the base edge in meters?
A. 10
C. 12
B. 11
D. 13
C. 1210.56
D. 2316.75
27. If the edge of a cube is increased by 30%, by how much is the
surface increased?
A. 30%
B. 21%
C. 69%
D. 33%
40. The volume a regular pyramid whose base is a regular hexagon is
156 m3. If the altitude of the pyramid is 5 m., find the sides of the
base.
A. 4 m
B. 8 m
C. 6 m
D. 3 m
28. Compute the volume of a regular icosahedron with sides equal
to 6 cm.
A. 470.88 cm3
B. 520.78 cm3
C. 340.89 cm3
D. 250.56 cm3
41. If the surface areas of two spheres are 24 cm2 and 96 cm2
respectively. Find the ratio of their volume.
A. ¼
B. 5
C. 1/8
D. 3/5
29. What is the shape of the base of a hexagonal prism?
A. hexagon
B. rectangle
C. pentagon
D. square
42. If the edge of a cube is doubled, which of the following is
incorrect?
A. The lateral area will be quadrupled
B. The volume is increased 8 times
C. The diagonal is doubled
D. The weight is doubled
30. Which of the following relates the number of faces, vertices and
edges in a polyhedron?
A. Plato’s formula
B. Euler’s formula
C. Calma’s formula
D.Heron’s formula
31. It is a three-dimensional shape consisting of two parallel circular
bases, joined by a curved surface.
A. cone
B. prism
C. cylinder
D. frustum
32. Spherical balls 1.5 cm in diameter area packed in a box
measuring 6 cm by 3 cm by 3 cm. If as many balls as possible are
packed in the box, how much free space remains in the box?
A. 28.41 cc
B. 20.47 cc
C. 29.87 cc
D. 25.73 cc
33. The volume of the sphere is 3π cu.m. The surface area of this
sphere in sq. m. is:
A. 36 π
B. 24 π
C. 18 π
D. 12 π
34. Given two spheres whose combined volume is known to be 819
cu. m. if their radii are in the ratio 3:4, what is the volume of the
smaller sphere?
A. 576 cu. m.
B. 243 cu. m.
C. 343 cu. m.
D. 476 cu. m.
35. If the surface area of a sphere is increased by 21%, its volume is
increased by:
A. 13.31%
B. 33.1%
C. 21%
D. 30%
36. The base areas of a frustum of a cone are 25 sq. cm. and 16 sq.
cm, respectively. If its altitude is 6 cm, find its volume.
A. 120 cm3
B. 122 cm3
C. 129 cm3
D. 133 cm3
37. The slant height of a right circular cone is 5m long. The base
diameter is 6m. What is the lateral area in sq. m?
A. 37.7
B. 47
C. 44
D. 40.8
38. A frustum of a pyramid has an upper base 100 m by 10 m and a
lower base of 80 m by 8 m. if the altitude of the frustum is 5 m, find
its volume.
A. 4567.67 cu. m.
B. 3873.33 cu. m.
C. 4066.67 cu. m.
D. 2345.98 cu. m.
39. Find the volume of a spherical cone in a sphere of radius 17 cm if
the radius of its zone is 8 cm.
A. 2120.35
B. 1426.34
43. A reservoir is shaped like a square prism. If the area of its base is
225 sq. cm, how many liters of water will it hold?
A. 3.375
B. 3375
C. 33.75
D. 3375
44. The altitude of the frustum of a regular rectangular pyramid is
5m the volume is 140 cu. m. and the upper base is 3m by 4m. What
are the dimensions of the lower base in m?
A. 9 x 10
B. 6 x 8
C. 4.5 x 6
D. 7.50 x 10
45. The volume of a right circular cone is 36 π. If its altitude is 3, find
its radius.
A. 3
B. 4
C. 5
D. 6
46. A cubical container that measures 2 inches on each side is tightly
packed with 8 marbles and is filled with water. All 8 marbles are in
contact with the walls of the container and the adjacent marbles. All
of the marbles are of the same size. What is the volume of water in
the container?
A. 0.38 cu. in.
B. 2.5 cu. in.
C. 3.8 cu. in.
D. 4.2 cu. in.
47. How many polyhedrons are there in all?
A. 5
B. 6
C. 7
D. 8
48. A trough has an open top 0.30 m by 6 m and closed vertical ends
which are equilateral triangles 30 cm on each side. It is filled with
water to half its depth. Find the volume of the water in cubic
meters.
A. 0.058
B. 0.046
C. 0.037
D. 0.065
49. The capacities of two hemispherical tanks are in the ratio 64:125.
If 4.8 kg of paint is required to paint the outer surface of the smaller
tank, then how many kilograms of paint would be needed to paint
the outer surface of the larger tank?
A. 8.5 kg
B. 6.7 kg
C. 7.5 kg
D. 9.4 kg
50. Find the volume of a spherical segment of radius 10 m and the
altitude 5 m.
A. 654.5 cu. m.
B. 659.8 cu. m.
C. 675.2 cu. m.
D. 680.5 cu. m.
51. Which of the following statements is false?
A. The diagonals of a rhombus are perpendicular to each other.
B. The diagonals of a rectangle are equal.
C. The diagonals of a rhombus are equal.
D. The diagonals of a parallelogram bisect each other.
52. The sum of the interior angles of a convex polygon of n sides is
equal to how many right angles?
A. 2(n-1)
B. 2(n-2)
C. 2(n-3)
D.2(n-4)
53. The area of a circle is 6 times its circumference. What is its
radius?
A. 12
B. 11
C. 10
D. 13
54. If the length of a side of a square is increased by 100%, its
perimeter is increased by
A. 100 %
B. 150 %
C. 200 %
D. 250 %
55. Find the area of a triangle whose sides are 9, 12 and 15.
A. 54
B. 53
C. 52
D. 51
56. Each of four circles ( see figure ) is tangent to the other three. If
the radius of each of the smaller circles is 3, what is the radius of the
largest circle ?
A. 6.46
B. 6.64
C. 4.64
D.4.46
57. 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
58. If BC = 2(AB), what fraction of the circle is shaded?
A. 1/4
B. 1/3
C. 1/2
D. 1/5
59. A diagonal of a parallelogram is 56.38 ft long and makes an angle
of 26°13' and 16°24' respectively with the sides. Find the area of the
parallelogram.
A. 595 ft²
B. 585 ft²
C. 575 ft²
D. 565 ft²
60. A convex polygon is a polygon each interior angle of which is less
than
A. 45°
B. 60°
C. 180°
D. 90°
61. Find the common area of two intersecting circles of radii 12 and
18 respectively if their common chord is 14 long.
A. 34.19
B. 35.29
C. 36.39
D. 37.49
62. Compute the difference between the perimeters of a regular
pentagon and a regular hexagon if the area of each is 12.
A. 0.31
B. 0.21
C.0.41
D.0.51
63. 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.
A. 11
B. 12
C. 13
D. 14
64. If each interior angle of a regular polygon has a measure of 160°,
how many sides has the polygon ?
A. 16
B. 17
C. 19
D. 18
65. If in triangle ABC, A = 76°30', B = 81°40' and c = 368, find the
diameter of the circumscribed circle.
A. 989.5
B. 959.8
C. 395.8
D. 958.9
66. A circle whose radius is 10 cm is inscribed in a regular hexagon.
The area of the hexagon is
A. 346.4 cm²
B. 634.4 cm²
C. 364.4 cm²
D. 436.6 cm²
67. Find the area of a rhombus whose diagonals are 32 cm and 40
cm.
A. 540 cm²
B. 340 cm²
C. 640 cm²
D. 440 cm²
68. If the sum two exterior angles of a triangle is 230°, find the
measure of the third exterior angle.
A. 130°
B. 120°
C. 110°
D. 100°
69. The ratio of the sum of the exterior angles to the sum of the
interior angles of a polygon is 1:3. Identify the polygon.
A. hexagon
B. heptagon
C. octagon
D. nonagon
70. If the perimeter of a regular hexagon is 24, what is the apothem?
A. 3√3
B. 4√3
C. 2√3
D. 5√3
71. 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
72. If a regular polygon has 54 diagonals, then it has how many
sides?
A. 10
B. 11
C. 14
D. 12
73. If four angles of a pentagon have measures 100°, 96°, 87° and
97°, find the measure of the fifth angle.
A. 150°
B. 160°
C. 140°
D. 130°
74. The area of the triangle inscribed in a circle is 40 sq. cm. and the
radius of the circumscribed circle is 7 cm. If two sides of the triangle
are 8 cm and 10 cm, find the third side.
A. 10 cm
B. 12 cm
C. 13 cm
D. 14 cm
75. Which of the following points is equidistant from the vertices of
a triangle?
A. incenter
C. orthocenter
B. centroid
D. circumcenter
76. The circumference of two circles are 6π and 10π. What is the
ratio of their areas?
A. 9/25
B. 8/25
C. 7/25
D.6/25
77. 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
78. The area of a sector of a circle, having a central angle of 60° is 24
pi. Find the radius of the circle.
A. 11
B. 12
C. 13
D. 14
79. The sides of triangle ABC are a = 14, b = 12 and c = 10. Find the
length of the median from vertex A to side A.
A. 8.34
B. 8.44
C. 8.54
D.8.64
80. The minute hand of a large clock is 2 m long. Find the distance
traveled by the tip of the minute hand in 5 minutes.
A. pi/4
B. pi/2
C. pi/6
D. pi/3
81. The angles between the diagonals of a rectangle is 30° and each
diagonal is 12 cm long. Find the area of the rectangle.
A. 26 cm²
B. 36 cm²
C. 46 cm²
D. 56 cm²
82. How many sides has a convex polygon if the sum of the measure
of its angles is 1080°?
A. 8
B. 7
C. 6
D. 5
83. Find the altitude of a trapezoid of area 180 cm² if the bases are
16 and 14 cm.
A. 11 cm
B. 12 cm
C. 13 cm
D. 14 cm
84. What is the measure of each interior angle of a regular
pentagon?
A. 106°
B. 109°
C. 107°
D. 108°
85. The solid formed by revolving a circle about an external axis in its
plane is called.
A. annulus
B. conoid
C. torus
D. prismatoid
86. The intersection of two faces of a pyramid is called the
A. lateral edge
B. slant height
C. altitude
D. hypotenuse
87. What is the total area of a cube whose edge is 5 cm?
A. 150 cm²
B. 145 cm²
C. 140 cm²
D. 135 cm²
88. Find the volume of the frustum of a right circular cone whose
altitude is 6 and whose base radii are 2 and 3.
A. 35π
B. 36π
C. 37π
D. 38π
89. A sphere of radius R is inscribed in a cube of edge e. What is the
ratio of the volume of the sphere to the volume of the cube?
A. 0.6523
B. 0.5236
C. 0.3652
D. 0.2635
90. The slant height of a right circular cone is 13 and the altitude is
12. Find the radius of the base.
A. 8
B. 7
C. 6
D. 5
91. A hemispherical bowl of radius 10 cm is filled with water to a
depth of 5 cm. Find the volume of the water.
A. 615π/3 cm³
B. 620π/3 cm³
C. 625π/3 cm³
D. 630π/3 cm³
92. The volume of a sphere whose diameter is 20 cm is
A. 4198.87 cm³
B. 4179.88 cm³
C. 4188.79 cm³
D.4187.89 cm³
93. It is a polyhedron whose faces are all squares.
A. tetrahedron
B. octahedron
C. hexahedron
D. dodecahedron
94. Find the length of the diagonal of a rectangular box whose edges
are 6, 8 and 10.
A. 7√2
B. 8√2
C. 9√2
D. 10√2
95. Find the area of the base of a prism whose volume is 516.6 cu. ft
and whose height is 16.4 in.
A. 372 ft²
B. 374 ft²
C. 376 ft²
D. 378 ft²
96. What is the volume of a pyramid whose altitude is 27 and whose
base is a square 8 on a side?
A. 756
B. 657
C. 576
D. 675
97. The base radii of the frustum of a cone are 6 cm and 10 cm
respectively. Find the altitude of the frustum if its volume is 1176π
cu. cm?
A. 16 cm
B. 17 cm
C. 18 cm
D. 19 cm
98. What is the diameter of a sphere for which its volume is equal to
its surface area?
A. 7
B. 6
C. 5
D. 4
99. Find the volume of a right circular cone whose base radius is 8
cm and whose altitude is 15 cm.
A. 320 pi cm³
B. 330 pi cm³
C. 340 pi cm³
D. 350 pi cm³
100. It is a polyhedron of which one face is a polygon and the other
faces are triangles which have a common vertex.
A. prism
B. pyramid
C. cone
D. prismatoid