# Trigonometry and Solid Mensuration Additional Exercises ```ADDITIONAL PRACTICE EXERCISES in
TRIGONOMETRY and SOLID
MENSURATION
1. The sides of a triangular lot are130 m, 180 m and
190 m. This lot is to be divided by a line bisecting the
longest side and drawn from the opposite vertex.
Find the length of this line.
A. 120 m
C. 122 m
B. 130 m
D. 125 m
2. The areas of two similar triangles are in the ratio of
4:1. If the length of the side of the smaller triangle is
5 units, what is the length of the corresponding side
of the larger triangle?
A. 15
C. 10
B. 25
D. 20
3. Sin (B – A) is equal to _____, when B = 270 degrees
and A is an acute angle.
A. –cos A
C. –sin A
B. cos A
D. sin A
4. If sec² A is 5/2, the quantity 1 - sin² A is equivalent to
A. 2.5
C. 0.4
B. 1.5
D. 0.6
5. A man standing on a 48.5 ,meter building high, has
an eyesight height of 1.5 m from the top of the
building, took a depression reading from the top of
another nearby building and nearest wall, which are
50° and 80° respectively. Find the height of the
nearby building in meters. The man is standing at
the edge of the building and both buildings lie on the
same horizontal plane.
A. 39.49
C. 30.74
B. 35.50
D. 42.55
6. Solve for θ in the following equation: Sin 2θ = cos θ
A. 30°
C. 60°
B. 45°
D. 15°
7. Two triangles have equal bases. The altitude of one
triangle is 3 units more than its base and the altitude
of the other triangle is 3 units less than its base. Find
the altitudes, if the areas of the triangles differ by 21
square units.
A.6 and 12
C. 5 and 11
B.3 and 9
D. 4 and 10
8. A ship started sailing S 42°35’ W at the rate of 5 kph.
After 2 hours, ship B started at the same port going
N 46°20’ W at the rate of 7 kph. After how many
hours will the second ship be exactly north of ship
A?
A.3.68
C. 5.12
B. 4.03
D. 4.83
9. An aero lift airplane can fly at an airspeed of 300
mph. If there is a wind blowing towards the cast at
50 mph, what should be the plane’s compass
heading in order for its course to be 30°? What will
be the plane’s ground speed if it flies in this course?
A. 19.7°,307.4 mph
B. 20.1°,309.4 mph
C. 21.7°,321.8 mph
D. 22.3°,319.2 mph
10. If the vertex of an isosceles triangle is 80  and the
side opposite the vertex measures 12 cm, determine
the perimeter of the triangle.
A. 30.66 cm
C. 34.88 cm
B. 32.53 cm
D. 28.55 cm
11. The sides of a scalene triangle are 5, 7 and 10 cm
respectively. Determine the radius of the inscribed
circle.
A. 1.48
B. 2.05
C. 2.22
D. 1.53
12. The sides of a triangle ABC are AB = 15 cm, BC =
18 cm and CA = 24 cm. Determine the distance from
the point of intersection of the angular bisector to
side AB.
A. 5.21 cm
C. 3.78 cm
B. 4.73 cm
D. 6.25 cm
13. A ladder 5 m long leans on a wall and makes an
angle of 30° with the horizontal. Find the vertical
height from the top to the ground.
A. 2.50 m
C. 1.50 m
B. 2.00 m
D. 2.75 m
14. If sin 3A = cos 6B, then:
A. A  B  180
C. A  2B  30
A

2B

30

B.
D. A  B  30
15. The sides of a triangle are 14 cm, 15 cm and 13 cm
respectively. Find the area of the circumscribing
circle.
A. 202.2
C. 210.5
B. 205.3
D. 207.4
16. The two legs of a triangle are 300 and 150 m each,
respectively. The angle opposite the 150 m side is
26°. What is the third side?
A. 197.49 m
C. 341.78 m
B. 218.61 m
D. 282.15 m
17. If sin A = 4/5 and A is in the first quadrant, find tan A
+ sec A.
A. 3
C. 4
B. 3/5
D. 2/3
18. If sin   cos   1/ 3 , what is the value of sin 2 ?
A. 1/3
C. 8/9
B. 1/9
D. 4/9
19. Determine the simplified form of 2/(1-cos 2C).
A. csc C
C. se C
B. sec2 C
D. csc2 C
20. Points A and B are 100 m apart and are of the same
elevation as the foot of a building. The angles of
elevation of the top of the building from points A and
B are 21° and 32° respectively. How far is A from the
building in meters?
A. 259.28
C. 271.64
B. 265.42
D. 277.29
21. A PLDT tower and a monument stand on a level
plane. The angles of depression of the top and
bottom of the monument viewed from the top of the
PLDT tower at 13° and 35° respectively. The height
of the tower is 50 m. Find the height of the
monument .
A. 29.13 m
C. 32.12 m
B. 30.11 m
D. 33.51 m
22. The captain of a ship views the top of a lighthouse at
an angle of 60° with the horizontal at an elevation of
6 meters above sea level. Five minutes later, the
same captain of the ship views the top of the same
lighthouse at an angle of 30° with the horizontal.
Determine the speed of the ship if the lighthouse is
known to be 50 meters above sea level.
A. 0.265 m/sec C. 0.169 m/sec
B. 0.155 m/sec D. 0.021 m/sec
23. If an equilateral triangle is circumscribed about a
circle of radius 10 cm, determine the side of the
triangle.
A. 34.64 cm
C. 36.44 cm
B. 64.12 cm
D. 32.10 cm
24. A wheel, 5 ft in diameter, rolls up an incline of 18°20’.
What is the height of the center of the wheel above
the base of the incline when the wheel has rolled up
5 ft up the incline?
A. 3 ft
C. 4 ft
B. 5 ft
D. 6 ft
25. ( cos A )4 – ( sin A ) 4 is equal to _____.
A. cos 4A
C. sin 2A
B. cos 2A
D. sin 4A
26. Csc 520° is equal to
A. cos 20°
C. tan 45°
B. csc 20°
D. sin 20°
27. Find the value of A between 270° and 360° if 2 sin2
A – sin A = 1
A. 300°
C. 310°
B. 320°
D. 330°
28. If cos 65° + cos 55° = cos θ, find θ in radians.
A. 0.765
C. 1.213
B. 0.087
D. 1.421
29. The area of a triangle inscribed in a circle having a
radius 9 cm. is equal to 43.23 sq. cm. If one of the
sides of the triangle is 18 cm., find one of the other
sides.
A. 16.42 cm
C.14.36cm
B. 17.29 cm
D. 12.42 cm
30. The sine of a certain angle is 0.6 calculate the
cotangent of the angle.
A. 4/3
C. 4/5
B. 5/4
D. 3/4
31. Find the value of y in the given: y = (1 + cos 2θ) tan
θ
A. sin θ
C. sin 2θ
B. cos θ
D. cos 2θ
32. Simplify the equation sin2 θ ( 1+cot2 θ )
A.1
C. sin2 θ sec2 θ
B. sin2 θ
D. sec2 θ
33. Simplify the expression sec θ – (sec θ )sin2 θ
A. cos2 θ
C. sin2 θ
B. cos θ
D. sin θ
34. Evaluate arc cot [ 2 cos )arc sin 0.5) ]
A. 30°
C. 60°
B. 45°
D. 90°
35. Solve for the x in the given equation: arc tan (2x) +
arc tan (x) = π/4
A. 0.149
C. 0.421
B. 0.281
D.0.316
36. A man finds the angle of elevation of the top of a
tower to be 30°. He walks 85 m nearer the tower and
finds its angle of elevation to be 60°. What is the
height of the tower?
A. 76.31 m
C. 73.31 m
B. 73.16 m
D. 73.61 m
37. Solve for x in the equation:
A. 90°
C.95°
B.100°
D. 80°
sin2x  2cosx .
38. If sin 40  sin20  sin  , find the value of θ in degrees.
A. 20
C. 120
B.80
D. 60
39. From A a pilot flies 125 km in the direction of
N3820’W and turns back. Through an error, the
pilot then flies 125 km in the direction of S5140’E.
In what direction must the pilot now fly to reach the
intended destination A?
A. S 4220’ W
C. S 4420’ W
B. S 4320’ W
D. S 4520’ W
40. Evaluate:
A. 1.091
B. 0.021
sec(arc csc 5 / 2) .
C. 2.191
D. 1.001
41. A pole is erected on the top of a building 20 m high.
The angles of elevation of the foot and the top of the
pole from a point on the ground are 30  and 40 
respectively. Find the height of the pole.
A. 8.87 m
C. 7.75 m
B. 9.07 m
D. 8.24 m
42. The angle or inclination of ascend of a road having
A. 4.72
C. 5.12
B. 4.27
D. 1.86
43. Given the 3 sides of a triangle: 2, 3, 4. What is the
angle in radians opposite the side with length 3?
A. 0.81
C. 0.67
B. 0.45
D. 0.74
44. In triangle DEF, DE = 18m and EF = 6m. Side FD
maybe
A. 10 m
C. 12 m
B. 11 m
D. 13 m
45. Given an equilateral triangle. What is the ratio of the
area of a circle inscribed to the area of a circle
A. 1/2
C. 1/3
B. 1/4
D. 1/5
50. The angle of elevation of a tower at place A south of
it is 30  and at B west of A and a distance of 50 m
from it the angle of elevation is 18  , determine the
height of the tower.
A. 19.65 m
C. 22.22 m
B. 15.42 m
D. 26.32 m
51. A flagpole stands on the edge of the bank of a river.
From a point on the opposite bank directly across
from the flagpole, the measure of the angle of
elevation to the top of the pole is 25. From a point
200 ft further away and in line with the pole and the
first point, the measure of the angle of elevation to
the top of the pole is 21. Find the distance across
the river.
A. 930 ft
C. 940 ft
B. 950 ft
D. 960 ft
46. In a triangle ABC, AB = 36 cm, BC = 24 cm and CA
= 18 cm. Find the length of the median from the
vertex C to the line AB.
A. 23.12
C.11.22
B. 16.78
D. 20.12
52. The bearing of a buoy from a ship 8.7 mi away is 64.
The ship is headed due north and the navigator
plans to change course when the buoy has a bearing
of 154. How much farther will the ship travel before
a change of course is needed?
A. 18.5 mi
C. 19.8 mi
B. 17.3 mi
D. 20.4 mi
47. In an oblique triangle, a = 25, b = 16, angle C =
94°06’. Find the measure of angle A.
A. 54.5°
C. 45.5°
B. 24.5°
D. 55.4°
53. Find the exact values cos (α+β) if sin α = 3/5, tan β
= -5/12 , 0<α<π/2 and π/2<β<π.
A. 33/65
C. -33/65
B. -63/55
D. 63/65
48. sin A cos B – cos A sin B is equivalent to :
A. cos ( A – B ) C. tan ( A – B )
B. sin ( A – B )
D. cos ( A
–B)
49. A parcel of land in the form of a triangle has sides
312 m and 485 m long, respectively. The included
angle is 81  30’. Determine the perimeter of the land.
A. 1243.22 m
C. 1333.5 m
B. 1354.76 m
D. 1426.3 m
54. One of the angles of a triangle measures 31. An
adjacent side measures 40 cm and the opposite side
is twice as long as the other adjacent side. What are
the measures of the other two angles?
A. 13 and 136 C. 14 and 135
B. 16 and 133 D. 15 and 134
55. A space-shuttle pilot flying toward the Suez Canal
finds that the angle of depression to one end of the
canal is 38.25 and the angle of depression to the
other end is 52.75. If the canal is 100.6 mi long, find
the altitude of the space shuttle.
A. 196 miles
C. 198 miles
B. 192 miles
D. 194 miles
56. Three circles with centers A,B, and C have
respective radii 50, 30 and 20 in and are tangent to
each other externally. Find the area (in in2) of the
curvilinear triangle formed by the three circles.
A. 142
C. 146
B. 152
D. 148
57. Three circles of radii 115,150 and 225m,
respectively, are tangent to each other externally.
Find the smallest angle of the triangle formed by
joining the centers of the circles.
A. 33.5
C. 61.2
B. 43.1
D. 51.9
58. A woman hikes 503 m, turns and jogs 415 m, turns
again, and runs 365 m returning to her starting point.
What is the area (in m2) of the triangle formed by her
path?
A. 77,800
C. 76,200
B. 75,400
D. 74,600
59. If
sin  
2
2
,
tan   1 ,

   ,
2
and
3
   2 ,
2
find
the exact value of sin(α+β).
A. 0
C. 1
B. -2
D. -1
60. A tower 150 m high is situated at the top of a hill. At
a point 650 m down the hill the angle between the
surface of the hill and the line of sight to the top of
the tower is 1230’. Find the inclination of the hill to
the horizontal plane.
A. 554’
C. 612’
B. 710’
D. 750’
61. Find the length of the altitude of angle A to side a for
a triangle whose sides are a = 16, b = 14, and c =
20.
A. 14
C. 13
B. 15
D. 12
62. Two forces of 17.5 and 22.5 lb act on a body. If their
directions make an angle of 5010’ with each other,
find the magnitude of their resultant.
A. 36.3 lb
C. 35.6 lb
B. 38.6 lb
D. 37.3 lb
63. A pole casts a shadow of 25 ft at one time and a
shadow of 10 ft at a later time when the angle of
elevation is twice as large. Find the height of the
pole.
A. 10 ft
C. 11 ft
B. 12 ft
D. 13 ft
64. Simplify: sin2 x 1 cot 2 x  .
A. sin2 x
B. sec2 x sin x
C.1
D. tan2 x
65. What is the area, to the nearest acre, of a triangular
field that is 529 ft on one side and 849 ft on another,
if the angle between these sides measures 102.7 ?
A. 7
C. 6
B. 4
D. 5
66. Find the identity of sin3θ in terms of sinθ.
A. 3 sin   4 sin3 
C. 4 sin   3 sin3 
B. 4 sin   3 sin3 
D. 3 sin   4 sin3 
67. A parcel of land has sides measuring 175 ft, 234 ft,
295 ft and 415 ft and the angle between the sides of
length 234 ft and 295 ft has measure 137.1. What
is the measure of the angle opposite this angle?
A. 110
C. 108
B. 104
D. 106
68. Solve for A (0⁰  A  2 A) in the equation 2sin A
cos A – sin A – 2cos A + 1 = 0.
A. A/6
C. A/4
B. A/3
D. 2 A/3
74. If A + B + C = 180⁰ and tan A + tan B + tan C = x,
find the value of (tan A)(tan B)(tan C).
A. 1.50x
C. 2.0x
B. 0.50x
D. x
69. In the system of equations below, find the average
of x and y.
sin x + sin y = a ; cos x + cos y = b
A. Arcsin (a/b)
C. Arctan (a/b)
Arccot ((a2 +
B. Arccos(a/b)
D.
b2)/2)
75. The explement of an angle is 5 times its supplement.
Find the angle.
A. 135⁰
C. 45⁰
B. 150⁰
D. 90⁰
70. Two chords AB = 8 cm and BC = 12 cm of a circle
form an angle of 120⁰. Find the radius of the circle.
A. 14 cm
C. 12 cm
B. 16 cm
D. 10 cm
71. A quadrilateral ABCD is to be inscribed in a
semicircle in such a way that AD will be along the
diameter. If AB = 10 cm, BC = 20 cm, and CD = 30
cm, determine the diameter of the semicircle.
A. 41.1309 cm
C. 41.1903 cm
B. 41.1039 cm
D. 41.1093 cm
72. At a point A south of a tower, the angle of elevation
of the top of the tower is 30⁰. From A, 1000 meters
east, at point B, the angle of elevation of the top of
the tower is 15⁰. Find the height of the tower.
A. 320.05 m
C. 230.50 m
B. 302.50 m
D. 203.05 m
73. Two towers 120 meters apart are on the same
horizontal plane. An observer standing successively
at the bases of the towers observes that the angle of
elevation of the taller is twice that of the smaller. At
the midway point, he observes that the angles of
elevation of the tops of the towers are
complementary. Find the height of the smaller tower.
A. 90 m
C. 40 m
B. 80m
D. 30 m
76. Find the value of sin2 0⁰ + sin2 1⁰ +…+ sin2 89⁰ + sin2
90⁰.
A. 44.5
C. 46.5
B. 45.5
D. 47.5
77. Of what quadrant is A if sec A is positive and csc A
is negative?
A. III
C. IV
B. I
D. II
78. If cos 4x = cos 6y, which of the following is correct?
A. 2x + 3y = 90⁰
C. 2x + 3y = 180⁰
B. 2x + 3y = 270⁰
D. 2x + 3y = 360⁰
79. In what direction should an airplane fly in order to fly
due north at 320 kph if the wind is blowing due west
at 50 kph
A. N 8⁰53’ W
C. S 8⁰53’ W
B. S 8⁰53’ E
D. N 8⁰53’ E
80. A baseball diamond is a square 27 m on a side, and
the pitcher’s mound is 18 m from the homeplate.
How far is the mound from the first base?
A. 19.12 m
C. 27.00 m
B. 20.13 m
D. 24.75 m
81. Find the angle between the diagonal of a cube and
the diagonal of a face of the cube. Both diagonals
are drawn from the same vertex.
A. 45.00⁰
C. 53.13⁰
B. 35.26⁰
D. 36.87⁰
82. A wheel 3.2 ft. in diameter rolls up an incline of 16⁰.
When the point of the contact of the wheel with the
incline is 4.5 feet from the base of the incline, how
far is the center of the wheel from the horizontal
base?
A. 3.67 ft
C. 2.78 ft
B. 4.98 ft
D. 2.37 ft
83. A vertical pole stands on a plane inclined at an angle
of 12⁰ with the horizontal. At the base of the incline
the pole subtends an angle of 30⁰. At a point from
the base and 6.2 meters up the incline, the pole
subtends an angle of 53⁰. Find the height of the pole.
A. 5.61 m
C. 7.89 m
B. 9.34 m
D. 6.48 m
84. An existing road (AT) is just passing along the
circumference of a circular lake where point T is the
point of tangency. A proposed road (AC) is to cross
the lake which requires a bridge (BC) and a detour
road (AB) to be constructed over the lake. If AT = 2.5
km and AB = 2.15 km, determine the length of the
bridge.
A. 0.76 km
B. 1.22 km
C.
D.
0.61 m
0.53 m
85. In the figure shown, determine the area of the
C
B

D
P
A
PB = 50 m, PC = 90 m, PA = 45 m, A = 15⁰
A. 981 m2
B. 874 m2
C.
D.
672 m2
701 m2
86. The bisector of the right angle of a right triangle
divides the hypotenuse into two segments, 431.9 cm
and 523.8 cm respectively. Find the one of the acute
angles of the given triangle.
A. 42.7⁰
C. 39.5⁰
B. 63.8⁰
D. 25.1⁰
87. An airplane is above a straight road on which are two
observers 1640 m apart. At a given signal the
observers take the angles of elevation of the
airplane, finding them to be 58⁰ and 63⁰,
respectively. Find the height of the plane.
A. 1378 m
C. 1298 m
B. 1503 m
D. 1443 m
88. Three times the sine of a certain angle is equal to
twice the square of the cosine of the same angle.
What is the tangent of the angle?
A. – 0.58
C. – 0.92
B. – 0.39
D. – 0.85
89. Two men A and B are 357 meters apart on a straight
shore of a lake and are looking at a tree T on the
opposite shore How wide is the lake if angle ABT is
32⁰ 20’ and angle BAT is 41⁰ 30’?
A. 182 m
C. 165 m
B. 132 m
D. 109 m
90. From the top of a tower 36 m high, the angles of
depression of the top and bottom of another tower
standing on the same horizontal plane are found to
be 28⁰ 56’ and 53⁰ 41’, respectively. Find the
distance between the summits of the two towers.
A. 25.9 m
C. 30.2 m
B. 24.3 m
D. 39.6 m
2
91. Simplify the equation sin x (1 + cot 2 x).
A. sin x
C. cot x
B. cos x
D. 1
92. A vertical pole consists of two parts, each one half of
the whole pole. At a point in the horizontal plane
which passes through the foot of the pole and 36 m
from it, the upper half of the pole subtends an angle
whose tangent is 1/3. How high is the pole?
A. 36
C. 24
B. 48
D. 18
93. ABDE is a square section and BDC is an equilateral
triangle with C outside the square. Compute the
angle of ACE.
A. 45⁰
C. 60⁰
B. 30⁰
D. 90⁰
94. The angle of elevation of the top of a tower from a
point A is 23⁰ 30’. From another point B, the angle of
elevation of the top of the tower is 55⁰ 30’. The points
A and B are 217.45 m apart and on the same
horizontal plane as the foot of the tower. The
horizontal angle subtended by A and B at the foot of
the tower is 90⁰. Find the height of the tower.
A. 92.15 m
C. 90.59 m
B. 86.34 m
D. 84.36 m
95. A tree broken over by the wind forms a right triangle
with the ground. If the broken part makes an angle
of 50⁰ with the ground and the top of the tree is now
20 feet from its base, how tall was the tree?
A.
C.
B.
D. 54.95 m
96. A certain road has a grade of 40%. What is its angle
of inclination in mils?
A. 388
C. 384
B. 390
D. 396
97. A cyclic quadrilateral has the sides AB = 8, BC = 10,
CD = 12 cm, respectively. The fourth side DA is the
diameter of the circle. Find the area of the circle.
A. 319.21 cm2
C. 321.34 cm2
2
B. 314.16 cm
D. 308.76 cm2
98. The vertices A and B of a quadrilateral lie on a circle
and are collinear with an external point P. A secant
is drawn to the circle intersecting at C and D. Angle
BPC = 30 degrees. If secants PA and PD have
lengths of 100 m and 90 m, respectively and the
external segment PB is 50 m, determine the area of
A. 1565.65 m2
C. 1666.67 m2
B. 1656.56 m2
D. 1555.56 m2
B. 14.22 cm
99. A border is formed by two concentric circles of
different radii. If the length of the chord of the larger
circle that is tangent to the smaller circle is 40, find
the area of the border.
A. 1,471.23
C. 1,256.6
B. 628.32
D. 859.31
100. Find the area of a regular pentagon whose side
is 25 cm and apothem is 7.2 cm.
A. 1075
C. 1142
B. 1010
D. 1067
101. Find the sum of the interior angles of
pentagram (star).
A. 180
C. 540
B. 360
D.720
the
102. A circle with radius 6 has half its area removed
by cutting of a border of uniform width. Find the width
of the border.
A. 2.20
C. 1.86
B. 1.76
D. 2.75
103. What is the measure of the interior angle of the
regular 2000-gon?
A. 1800
C. 2000
0
B. 220
D. 1780
104. For a regular polygon of heptagon sides, find the
number of
the degrees contained in each central
angle.
A. 51.43 0
C. 600
0
B. 400
D. 32.730
D. 14.23 cm
107. Find the length of the side of a regular pentagon
inscribed in a circle of a radius 10 cm.
A. 10.34 cm
C.11.76cm
B. 12.42 cm
D. 35.22 cm
108. The interior angles of a polygon are in arithmetic
progression. The least angle is 1200 and the
common deference is 50. Find the numbers of sides.
A. 8
C. 9
B. 11
D. 10
109. One side of a parallelogram is 10 cm and its
diagonals are 16 cm and 24 cm respectively. Find its
area.
A. 143.2 cm 2
C. 112.4cm2
2
B. 150.5cm
D. 158.74 cm2
110. The sum of the interior angles of a polygon is 540
degrees. Find the number of sides.
A. 5
C. 8
B. 6
D.11
111. The lengths of the diagonals of a rhombus are 3
cm and 4 cm, respectively. Find the perimeter of the
rhombus.
A. 9
B. 10 C. 11 D. 12
112. The ratio of the area of regular polygon inscribed
in a circle to the area of the circumscribing regular
polygon of the same number of side is 3:4. Find the
number of the sides.
A. 6
B. 8 C. 7 D. 9
105. Determine the area of a rhombus if each side
measure 2cm.
A. 2 2
C. 3 3
B. 2 3
D. 3 2
113. A rectangular ABCD which measures 18 by24
units is folded once, perpendicular to diagonal AC,
so that the opposite vertices A and C coincide. Find
the length of the fold.
A. 45/2
C. 7/2
B. 2
D. 54/2
106. The area of a rhombus is 168m2. If one of its
diagonal is12m, find the length of the sides of a
rhombus.
A. 15.23 cm
C.10.42cm
114. The diameters of the two circles that are tangent
internally are 18 and 8, respectively. What is the
length of the tangent segment from the center of the
larger circle to the smaller circle?
A. 2
C. 3
B. 4
D. 5
115. A regular 5 pointed star is inscribed in a circle
with a diameter of 10 m. What is the area of the
region inside the circle not covered by the star
A. 50.46 m2
C. 65.34 m2
B. 40.56 m2
D. 57.83 m2
116. A regular pentagon has sides of 20 cm. An inner
pentagon with sides of 10 cm is inside and
concentric to the larger pentagon. Determine the
area inside the larger pentagon but outside of the
smaller pentagon.
A. 515.76
C. 416.57
B. 637.15
D. 218.96
117. A quadrilateral is inscribed in a circle having a
is the diameter and the other two sides adjacent to
the diameter are 8 cm and 12 cm respectively.
Determine the length of the fourth side.
A. 8.63 cm
C. 10.33 cm
B. 9.86 cm
D. 9.42 cm
118. Which plane figure has the highest ratio of
perimeter to area?
A. circle
C. regular hexagon
B. equilateral triangle
D. square
119. A goat is tethered to a corner of a 4 m. by 5 m.
shed by a 6 m. rope. What is the maximum area the
goat can cover? (ans. in square meters)
A. 88.75
C. 67.87
B. 98.76
D. 56.48
120. If the sides of a parallelogram and an included
angle are 6, 10, and 100 degrees respectively, find
the length of the shorter diagonal.
A.10.63
C. 10.73
B.10.37
D. 10.23
121. The sides of a parallelogram are 84 cm and 63
cm respectively. If one of its diagonal is 48 cm long,
compute the smallest interior angle of the
parallelogram.
A. 50.64°
C. 45.43°
B. 36.42°
D. 34.51°
128. How many diagonals can be drawn from a 12
sided polygon?
A. 66
C. 54
B. 48
D. 36
122. The sum of the sides of two polygons is 9 and
the sum of its diagonals is 7. Find the number of
sides of each polygon.
A. 5 & 4
C. 7 & 2
B. 6 & 3
D. 4 & 8
129. The tangent and a secant are to a circle from the
same external point. If the tangent is 6 inches and
the external segment of the secant is 3 inches,
compute the length of the secant.
A. 10
C. 12
B. 13
D. 14
123. Tangents are drawn to a circle of radius 10 cm
from a point 25 cm from its center. Find the length of
the tangents.
A. 22.913 cm
C. 24.302 cm
B. 20.633 cm
D. 27.439 cm
124. Find the radius of the circle circumscribing an
isosceles right triangle having an area of 162 sq.
cm.?
A. 11.23
C. 12.73
B. 10.63
D. 15.27
125. The sides of the triangle are 40 cm, 50 cm, and
60 cm. How far is the point of intersection of the
perpendicular bisector of the sides of the triangle to
any of its vertex?
A. 30.24
C. 23.54
B. 15.27
D. 40.13
126. A triangle has sides equal to 68 cm, 77 cm, and
75 cm, respectively. Find the area of the escribed
circle tangent to the shortest side of the triangle.
A. 9,503.32 cm2
B. 6,503.23 cm2
C. 9,305.32 cm2
D. 8,603.32 cm2
127. A quadrilateral has sides equal to 12 cm, 20 cm,
8 cm, and 17 cm respectively. If the sum of the two
opposite angles is 225, find the area of the
A. 168.18
C. 70.73
B. 78.31
D. 186.71
130. 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 tangents?
A. 7.8 m
C. 10.7 m
B. 9.8 m
D. 6.7 m
131. 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
C. 7.5 kg
B. 6.7 kg
D. 9.4 kg
132. A wooden cone of altitude 10 cm is to be cut into
two parts of equal weight. How far from the vertex
should the cut parallel to the base be made?
A. 6.65 cm
C. 7.94 cm
B. 3.83 cm
D. 8.83 cm
133. A sphere of radius 5 cm and a right circular cone
of base radius 5 cm and a height 10 cm stand on a
plane. Find the position of a plane that cuts the two
solids in equal circular sections.
A. 2cm
C. 1.5 cm
B. 2.5 cm
D. 3.2 cm
134. 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
135. Two cylinders of equal radius 3 m have their
axes at right angles. Find the volume of the common
part.
A. 122 cu. cm.
C. 154 cu. cm.
B. 144 cu. cm.
D. 134 cu. cm.
136. A solid has a circular base of radius 20 cm. Find
the volume of the solid if every plane section
perpendicular to a certain diameter is an equilateral
triangle.
A. 18,475.21 cm3
C. 12,775.21 cm3
3
B. 20,475.31 cm D. 21,475.21 cm3
137. If the edge of a cube is increased by 30%, by
how much is the surface area increased?
A. 30%
C. 69%
B. 21%
D. 33%
138. If the edge of a cube decreases by x%, its
volume decreases by 48.8%. Find the value of x.
A. 10
B. 20
C. 16
D. 25
139. Compute the volume of a regular icosahedron
with sides equal to 6 cm.
A. 470.88 cm3
C. 340.89 cm3
3
B. 520.78 cm
D. 250.56 cm3
140. Compute the volume (in cm3)of a sphere
inscribed in an octahedron having sides equal to 18
cm.
A. 1622.33
C.1663.22
B. 1875.45
D.1892.63
141. Compute the area of bi-rectangular spherical
triangle having an angle of 60o and a radius of 8 m.
A. 76.56 m2
C. 56.34 m2
B. 45.65 m2
D. 67.02 m2
142. 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
C. 1210.56
B. 1426.34
D. 2316.75
143. A spherical wooden ball 15 cm in diameter sinks
to a depth of 12 cm in a certain liquid. Calculate the
area exposed above the liquid in cm2.
A. 45 pi.
C. 15 pi.
B. 20 pi.
D. 10 pi.
144. Two balls, one 6 cm in diameter and the other 4
cm in diameter are placed in a cylindrical jar 9 cm in
diameter. Find the volume of water necessary to
cover them.
A. 234.44 cm3
C. 362.33 cm3
3
B. 257.22 cm
D. 262.22 cm3
145. Find the area of a regular polygon whose side is
25 m and apothem is 17.2 m.
A. 1075
C. 1175
B. 925
D. 1275
146. Find the area of a pentagon which is
circumscribing a circle having an area of 420.60 sq.
cm.
A. 386.57
C. 486.35
B. 450.54
D. 260.24
147. A reverse curve on a railroad track consists of
two circular arcs. The central angle of one side is 20°
with radius of 2500 ft, while the central angle of the
other is 25° with radius 3000ft. Find the total lengths
of the two arcs.
A. 2182 ft
C. 2218 ft
B. 2812 ft
D. 2128 ft
148. Two cylinders of equal radius 3 m have their
axes at right angles. Find the volume of the common
part.
A. 144 cm3
C. 136 cm3
3
B. 155 cm
D. 125 cm3
149. Given a solid right circular cone having a height
of 8 cm. has a volume equal to 4 times the volume
of the smaller cone that could be cut from the same
cone having the same axis. Compute the height of
the smaller cone.
A. 5.04 cm
C. 4.45 cm
B. 3.25 cm
D. 2.32 cm
150. The diameter of a sphere and the base of a cone
are equal. What percentage of that diameter must
the cones height be so that both volumes are equal.
A. 100%
C. 200%
`
B. 50%
D. 400%
151. The volume of a regular pyramid whose base is
a regular hexagon is 156 m3. If the altitude of the
pyramid is 5m., find the sides of the base.
A. 4 m
C. 6 m
B. 8 m
D. 3 m
152. What is the volume of a pyramid whose altitude
is 16 cm. long and whose base is enclosed by a
rhombus whose sides are 6 cm. long and whose
acute angles are 30o?
A. 96 cm3
C. 86 cm3
3
B. 69 cm
D. 76 cm3
153. The base of a cylinder is a hexagon inscribed in
a circle. If the difference in the circumference of the
circle and the perimeter of the hexagon is 4 cm., find
the volume of the cylinder if it has an altitude of 20
cm.
A. 10367 cm3
C. 10123 cm3
3
B. 12239 cm
D. 11231 cm3
154. A solid has a circular base of diameter 40 cm.
Find the volume of the solid if every section
perpendicular to a fixed diameter is an isosceles
right triangle.
A. 32223.56
C. 10666.67
B. 12555.60
D. 12044.48
155. The volume of a truncated prism with an
equilateral triangle as its horizontal base is equal to
3600 cm3. The vertical edges at each corners are 4,
6, and 8 cm., respectively. Find one side of the base.
A. 22.37
C. 37.22
B. 25.43
D. 17.89
156. Aluminum and lead have specific gravities of 2.5
and 16.48 respectively. If a cubical aluminum has
edge of 0.30 m., find the edge of a cubical block of
lead having the same weight as the aluminum.
A. 10 cm
C. 13 cm
B. 14 cm
D. 16 cm
157. How many edges are there in a regular
octahedron?
A. 8
C. 12
B. 20
D. 30
158. A hole 10 cm in diameter is to be punched out
from a right circular cone having a diameter of 16cm.
Determine the volume punched out if the height of
the cone is 24cm.
A.1599.06
C. 1099.56
B.1124.53
D. 1235.72
159. In a circle with a diameter of 10 meters, a regular
five pointed star touching its circumference is
inscribed. What is the area of the part not covered
by the star?
A. 60.42
C. 40.68
B. 40.58
D. 50.47
160. 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. 124.77 sq. cm C. 150.35 sq. cm.
B. 150.26 sq. cm. D. 130.77 sq. cm.
161. The area of an isosceles triangle is 36 m2 with
the smallest angle equal to one third of the other
angle. Find the length of the shortest side.
A. 12.98 m
C. 7.35
B. 5.73 m
D. 6.84
162. The volume of the cube is reduced to _______
If all the sides are halved.
A. 1/2 B. 1/8 C. ¼ D. 1/16
163. If the edge of the cube is decreased by 10%, by
what percent is the surface area decreases?
A. 19%
C. 89%
B. 81%
D. 10%
164. If the surface area of a sphere is increased by
30%, by what percent is the volume of the sphere
increased?
A. 51.8%
C.61.7%
B. 48.2%
D. 30%
165. The lateral faces of a square pyramid make an
angle 60ْ with the base. If the height of the pyramid
is 5 3 m, find its lateral area.
A. 200 m2
C. 320 m2
2
B. 120 m
D. 220 m2
166. Find the volume of a cone to be constructed from
a sector 50 cm in diameter and a central angle of
130ْ .
A. 1,990.43 cm3 C.1990.43 cm3
B. 1,490.34 cm3 D.1,670.26 cm3
167. A cone is inscribed in a hemisphere of radius r.
If the cone and the hemisphere share bases, find the
volume of the region inside the hemisphere but
outside the cone.
A. r 3 / 3
C. r 3
3
B. 7r / 3
D. 4r 3
174. A cone and the 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
C. 3
B. 2
D. 3
168. A cylinder is circumscribed about a right prism
having a square base one meter on an edge. The
volume of the cylinder is 6.283 cu. m. Compute its
altitude.
A. 3
C. 5.4
B. 4
D. 2.5
175. A room is 12 ft wide, 15 ft along 8 ft high. If an air
conditioner changes the air once every five minute,
how many cubic feet of air does it change per hour?
A. 17,280
C.19,553
B. 14,522
D, 16,733
169. The volume of a right prism is 234 cu. m. with an
altitude of 15 m. If the base of the prism is an
equilateral triangle, find the length of the base edge.
A. 5
C. 6
B. 7
D. 8
176. In a regular square pyramid, the length of each
side of the square base is 12 in and the length of
the altitude is 8 in. Find the length of the lateral edge.
A. 11.66 in
C.12.34in
D. 10.34 in
D. 15.21 in
170. The surface area of a zone of a spherical cone
having an altitude of zone of 2 cm. is equal to 201.06
sq. cm. Compute the volume of the spherical cone.
A. 1072.32 cm3 C. 1263.21 cm3
B. 2152.21 cm3 D. 1143.21 cm3
177. The space diagonal of a cube is 6. Find the
surface area of the cube.
A. 12
B. 56
C. 72
D. 36
171. What is the area of a lune whose angle is 85ْ on
a sphere of radius 30 cm.
A. 1,670.45 cm2 C. 2,570.53 cm2
B.2,670.35 cm2 D. 1,670.35 cm2
172. Find the area of a spherical triangle ABC,
A=125°, B=73°,C=84° in a sphere of radius 30 cm.
A. 1562.4 cm2
C. 1602.2 cm2
2
B.1567.3 cm
D. 1652.2 cm2
173. A lune has an area of 30 square meters. If the of
the lune is 90 degrees. What is the area of the
sphere?
A. 110 sq. m.
C. 120 sq. m.
B. 90 sq. m.
D. 150 sq. m.
178. The height of the 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?
A. 1:27
B. 8:27
C. 26:27
D. 24:27
179. A right circular cone has an altitude of 18 in and
a radius of 8 in and is 1 4 filled with water. Solve the
height of the water.
A. 10.33 in
C.11.62in
B. 11.34 in
D.10.67 in
180. A wooden pyramid is to be cut into 2 parts of
equal weights by a plane parallel to its base. Find
the ratio of the heights of the two parts.
A. 3.85
C. 4.65
B. 2.15
D. 3.12
C. 3.44m
181. If the edge of the cube is increased by 20 %, find
the percentage increase in volume?
A. 72.8%
C. 83.2 %
B. 76.3%
D. 63.3%
182. A metal cube of side 4 cm is melted and recast
into a rectangular block of dimensions 8 cm x 4 cm
x 2 cm. Find the percentage increase in the total
surface area.
A. 12%
C.18.33%
B. 16.67%
D. 21%
183. To what height above the earth’s surface must a
man be raised for him to see 1/4 of the earth’s
surface?
A. H = R
C. H = 2R
B. H = 1/2 R
D. H =3R
184. A certain volcanic eruption expelled 36 cubic
miles of lava into the atmosphere, losing 4,100 ft of
its original altitude. Assume the original volcano to
be a circular cone and the final volcano to be a
frustum. I f the material lost from the top of the
mountain contained 1 % of the erupted material, find
the angle of inclination of the volcano
A. 50.78ْ
C. 37.8ْ
B. 49.5ْ
D. 43.6ْ
185. 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 original
sphere.
A. 200 cm
B. 225 cm
C. 220 cm
D. 240 cm
186. 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. 3.14m
B. 3.24m
D. 3.54m
187. A right prism has a square base and a lateral
edge 10 cm. Find the volume if the lateral area is 120
cm2.
A. 60 cm3
C. 80 cm3
3
B. 70 cm
D. 90 cm3
188. A right section of an oblique prism is a hexagon
whose sides are 3cm, 4 cm, 4cm, 5cm, 6 cm and 7
cm. The measure of one lateral edge is 7 cm. Find
the lateral area. 203 m2
A. 203 cm2
C. 205 cm3
2
B. 207 cm
D. 209 cm3
189. A solid cylinder of radius 10” is inscribed in a
prism with equilateral triangular bases. Find the
volume of the portion of the prism that is outside the
cylinder if their common height is 100 inches.
A. 10454 in3
C. 20545 in3
3
B. 32655 in
D. 40475 in3
190. A sphere whose diameter is 12 feet is illuminated
by a point source of light 18 feet from the center of
the sphere. Find the area of the portion of the sphere
which is illuminated.
A. 48pi
C. 72pi
B. 60pi
D. 84 pi
191. A right circular cylinder has a base whose
diameter is 7 and height is 10. What is the surface
area of the cylinder, not including the bases?
A. 30pi
C. 70 pi
B. 50 pi
D. 90 pi
192. What are the dimensions of a solid cube whose
surface area is numerically equal to its volume?
A. 2
B. 4
C. 6
D. 8
193. A parallelogram is a quadrilateral in which
opposite sides are parallel and equal. A plane
figure with four straight sides.
194. Find the lateral area and the total surface area of
a right circular cylinder if the radius of its base
measures 5 in. and its altitude measures 8 in.
A. 110 pi
C. 130pi
B. 120 pi
D. 140pi
195. 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. 17280 ft3/hr
C. 16690 ft3/hr
3
B. 15460 ft /hr
D. 18720 ft3/hr
196. The base of a right prism, is an equilateral
triangle, each of whose sides measures 4 units. The
altitude of the prism measures 5 units. Find the
volume of the prism.
A. 45.45
C. 54.45
B. 34.64
D. 63.36
197. Find the volume of a regular square pyramid if
each edge of the base measures 10 in., and the slant
height of the pyramid measures 13 in.
A. 200
C. 800
B. 600
D. 400
198. Find the volume of a sphere whose radius
measures 7 in.
A. 1236.56
C. 1536.87
B. 1436.76
D. 1736.35
199. Find the volume of a right circular cylinder of
height 6 if it has the same lateral surface area as a
cube of edge 3.
A. 36pi
C. 18pi
B. 54pi
D. 72pi
200. Find the volume of a solid right circular cone
whose height is 4 ft and whose base has radius 3 ft.
A. 24pi
C. 64pi
B. 12pi
D. 48pi
201. What is the slant height of a solid right circular
cone whose height is 4 ft and whose base has radius
3 ft.
A. 5
C. 7
C. 6
D. 8
202. Find the surface area of a solid right circular
cone whose height is 4 ft and whose base has radius
3 ft.
A. 15pi
C. 25pi
B. 20pi
D. 30pi
203. A cone is generated by rotating a right triangle
with sides 3,4, and 5 about the leg whose measure
is 4. Find the total area and volume of the cone.
A. 12pi
C. 18 pi
B. 16 pi
D. 20 pi
204. A manufacturer wishes to change the container
in which his product is marketed. The new container
is to have the same volume as the old, which is a
right circular cylinder with altitude 6 and base radius
3. The new container is to be composed of a frustum
of a right circular cone so that the smaller bases of
the two frustums coincide. The radii of the bases of
the lower frustum are 3 ½ and 2, respectively, and
its altitude is 4. The radius of the upper base of the
upper frustum is 3. Find the altitude of the upper
frustum.
A. 2.8
C. 4.8
B. 3.6
D. 5.6
205. An isosceles triangle, each of whose base
angles is 27 and whose legs are 5.2, is rotated
through 180, using as an axis its altitude to the
base, find its volume.
A. 51
C. 55
B. 53
D. 57
206. How many glass paper weights each in the
shape of a dodecahedron 1 in. on an edge can be
manufactured by melting 1000 glass paper weights
each in the form of a cube 2 in. on an edge?
A. 945
C. 1546
B. 1043
D. 1873
```