Unit 3: Volume, Cross Sections, and Rotations
Section 3.1: Volume
Section Homework
Volume Formulas of Solids
Cube
Prism
Cylinder
V = s3
V = B⋅H
V = π ⋅ r2 ⋅ H
Cone
V=
1
⋅ π ⋅ r2 ⋅ H
3
s - side length
Sphere
V=
B - Area of base
4
⋅π ⋅ r3
3
Pyramid
V=
1
⋅B⋅H
3
Spherical Segment
V=
H - Height of solid
(
1
⋅ π ⋅ H 3 ⋅ r12 + 3 ⋅ r22 + H 2
6
)
r - Radius
Homework Exercises
1. Find the volume of a cube with the following side lengths:
(A) 5.2 m
(B) 3.7 in
(C) 0.8 km
2. Find the volume of a slab of stone measuring 3 m in length, 2 m in breadth and 25 cm in
thickness.
3. If the surface area of a cube is 96 sq cm, find its volume.
4. A tank contains 60,000 cu. m of water. If the length and breadth are 50 m and 40 m
respectively, find its depth.
5. If the volume of cube is 2197 cu cm, find the surface area of the cube.
6. A closed wooden box measures externally as 42 cm by 32 cm by 27 cm. The wood used is 1
cm thick. Find the internal capacity (volume) of the box.
7. A field is 600 m long and 50 m broad. A tank 30 m long, 20 m broad and 12 m deep is dug
in the field. The earth taken out of it is spread evenly over the field. Find the height of the
field raised by it.
8. What is the volume of the following triangular prism?
9. Find the volume of a closed right cylinder whose radius is 7 m and height is 10 m.
10. A hollow cylindrical tube, open at both ends is made of iron 1 cm thick. If the external
diameter is 12 cm and the length of tube is 70 cm, find the volume of iron used in making
the tube.
11. A field is 150 m long and 70 m broad. A circular tank of radius 5.6 m and depth 20 cm is
dug in the field and the earth taken out of it is spread evenly over the field. Find the height
of the field raised by it.
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Geometry
Unit 3: Volume, Cross Sections, and Rotations
Section 3.1: Volume
Section Homework
12. A cubic meter of iron is made into a wire of diameter 3.5 mm. Find the length of the wire.
13. A cylindrical bucket of diameter 28 cm and height 12 cm, is full of water. The water is
emptied into a rectangular tub of length 66 cm and breadth 28 cm. Find the height to
which water rises in the tub.
14. A rectangular piece of paper 33 cm long and 16 cm wide is rolled along its breadth to get a
cylinder of height 16 cm. Find the volume of the cylinder.
15. Find the volume of the following pyramids.
(A)
(B)
16. Find the volume of a cone, the radius of whose base is 5 m and whose height is 12 m?
17. The volume of a cone is 616 cubic meters. If the height of cone is 27 meters, find the
radius of its base.
18. Find the volume of a sphere of radius 2.1 cm.
19. Find the diameter of a sphere of volume 4851 cubic cm. What is the surface area of this
sphere?
20. The radius of an iron sphere is 3.5 cm. It is melted to form smaller spheres of diameter
1.75 cm. Find the number of smaller spheres formed.
21. Find the volume of a spherical segment having height 3 cm and radii .5 cm and 1 cm.
22. Nine tennis balls 3.5 cm in diameter are packed in three layers of three per layer into a
cylindrical can having radius r ≈ 3.771 in and height h ≈ 9.215 in . Nine tennis balls can
also be packed into a long cylindrical can with the balls in a straight line, having radius
r = 1.25 in and height h = 22.5 in .
(a) What is the total space wasted using each of the two packing methods, and which
wastes the most total space? (Space wasted = volume of can minus volume of balls
inside.)
(b) How much material (total surface area in square inches) is required for the two types of
cans used for packaging, and which one requires the most?
23. Which of the following water tanks will hold the most water when three-fourths full (in
terms of the height of the water in the tank)?
(a) A cylindrical tank with radius 20 ft and height 40 ft (height of water in tank = 30 ft);
2
total surface area given by S = 2π r + 2π rH .
(b) A spherical tank that has the same total surface area ( S = 4π r ) as the cylindrical tank
in (a) counting top and bottom. (Hint: First show that the radius r of the spherical
tank must be approximately 24.5 ft, then use the formula for a spherical segment
2
with
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h=
3
( 2r ) = 36.75 ft , r1 = 0 , and using the Pythagorean theorem to find r2 .)
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Geometry