Transcript 1-7 Three-Dimensional Figures

Volume of Prisms and
Cylinders
Section 9.4
Objectives:
• Find the volume of prisms and
cylinders.
Key Vocabulary
• Prism
• Cylinder
• Volume
Volume
• The volume of a solid is the number
of cubic units contained in its
interior.
Rectangular Prism
• A prism is a polyhedron
with two congruent faces,
called bases, that lie in
parallel planes.
• The other faces called
lateral faces, are
parallelograms formed by
connecting the
corresponding vertices of
the bases.
• The segments connecting
these vertices are lateral
edges.
Finding the Volume of a Prism
• The box shown is
5 units long, 3
units wide, and 4
units high.
• How many unit
cubes will fit in the
box?
• What is the
volume of the
box?
Finding the Volume of a Prism
• The base of the box is 5 units
by 3 units.
• This means 5 • 3, or 15 unit
cubes, will cover the base.
• Three more layers of 15 cubes
each can be placed on top of
the lower layer to fill the box.
• Because the box contains 4
layers with 15 cubes in each
layer, the box contains a total
of 4 •15 cubes, or 60 unit
cubes.
Conclusion - Volume of a Prism
• Because the box is completely
filled by the 60 cubes, and each
cube has a volume of 1 cubic unit,
it follows that the volume of the
box is 60 • 1, or 60 cubic units.
• The area of the base, 15 square
units, multiplied by the height of 4
units, yields the volume of the
box, 60 cubic units.
• So, the volume of the prism can
be found by multiplying the area
of the base by the height.
Volume of Prism
Base
h
w
𝓁
Area of base = B
Height of Prism = h
Volume of Prism = Area of base x height
= Bh
w
Volume of a Prism
Find the volume of the prism.
•
•
•
•
•
•
V = lwh
l = 8 in
w = 4 in
h = 1 ft = 12 in
V = (8)(4)(12)
V = 384 in3
8 in
1 ft
4 in
Example 1
Find the Volume of a Rectangular Prism
Find the volume of the box by determining
how many unit cubes fit in the box.
SOLUTION
The base is 5 units by 3 units. So, 3 · 5,
or 15 unit cubes are needed to cover
the base layer.
There are 4 layers. Each layer has
15 cubes. So, the total number of
cubes is 4 · 15, or 60.
ANSWER
The volume of the box is 60 cubic units.
Example 2
Find the Volume of a Prism
Find the volume of the prism.
b.
a.
SOLUTION
a. V = Bh
Write the formula for volume of a prism.
= (7 · 4) · 5
= 140
ANSWER
Area of rectangular base = l · w = 7 · 4.
Simplify.
The volume is 140 cubic inches.
Example 2
Find the Volume of a Prism
b. V = Bh
Write the formula for volume of a prism.
1–
= 2·8·6 ·3
Area of triangular base = – · 8 · 6.
= 72
Simplify.
ANSWER
1
2
The volume is 72 cubic feet.
Your Turn:
Find the volume of the prism.
1.
ANSWER
216 ft3
ANSWER
125 cm3
ANSWER
245 in.3
2.
3.
Cylinder
• A cylinder is a solid with
congruent circular bases that
lie in parallel planes.
• The altitude, or height of a
cylinder is the perpendicular
distance between its bases.
• The radius of the base is also
called the radius of the
cylinder.
• A cylinder is called a right
cylinder if the segment joining
the centers of the bases is
perpendicular to the bases.
Volume of Cylinder
r
r
Area Base = Area
of Circle = πr2
h
Formation of Cylinder
by adding up bases
Or
V = Bh
B = Area Base
h = height
V r
2
r = radius
h = height
h
Volume of a Cylinder
Find the volume of the cylinder.
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•
•
•
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V = πr2h
π = 3.14
r=6
h = 10
V = (3.14)(62)(10)
V = 1130.4 in3
6 in
10 in
Example 3
Compare Volumes of Cylinders
a. How do the radius and height of
the mug compare to the radius
and height of the dog bowl?
b. How many times greater is the
volume of the bowl than the
volume of the mug?
SOLUTION
a. The radius of the mug is 2 inches and the radius of
the dog bowl is 6 inches. The radius of the bowl is
three times the radius of the mug. The height of the
mug is the same as the height of the bowl.
Example 3
Compare Volumes of Cylinders
b. Volume of mug
V=
Write the formula for
volume.
πr2h
= π(22)(4)
= 16π
Volume of dog bowl
Substitute for r and for h.
Simplify.
V = πr2h
= π(62)(4)
= 144π
To compare the volume of the bowl to the volume of
the mug, divide the volume of the bowl by the
volume of the mug.
Volume of bowl
144π
–––––––––––––– = ––––– = 9
16π
Volume of mug
ANSWER
The volume of the bowl is nine times the
volume of the mug.
Your Turn:
Find the volume of the cylinder. Round your answer
to the nearest whole number.
1.
2.
ANSWER
38 ft3
ANSWER
16 in.3
ANSWER
126 m3
3.
Assignment
• Pg. 503 - 507, #1 – 57 odd