11.5 Explore Solids

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11.5 Explore Solids11.6 Volume of Prisms and
Cylinders

• You will identify solids
• You will find volumes of prisms and cylinders

• Essential Questions
• When is a solid a polyhedron?
• How do you find the volume of a right prism of
  right cylinder?

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Identify and name polyhedra
EXAMPLE 1
Tell whether the solid is a polyhedron. If it is,
name the polyhedron and find the number of faces,
vertices, and edges.
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Identify and name polyhedron
EXAMPLE 1
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Identify and name polyhedron
EXAMPLE 1
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for Example 1
GUIDED PRACTICE
Tell whether the solid is a polyhedron. If it is,
name the polyhedron and find the number of
faces, vertices, and edges.
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for Example 1
GUIDED PRACTICE
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for Example 1
GUIDED PRACTICE
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Use Eulers Theorem in a real-world situation
EXAMPLE 2
House Construction
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Use Eulers Theorem in a real-world situation
EXAMPLE 2
To find the number of vertices, notice that there
are 5 vertices around each pentagonal wall, and
there are no other vertices. So, the frame of the
house has 10 vertices.
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Use Eulers Theorem with Platonic solids
EXAMPLE 3
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EXAMPLE 4
Describe cross sections
Describe the shape formed by the intersection of
the plane and the cube.
SOLUTION
a. The cross section is a square.
b. The cross section is a rectangle.
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EXAMPLE 4
Describe cross sections
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for Examples 2, 3, and 4
GUIDED PRACTICE
4. Find the number of faces, vertices, and
edges of the regular dodecahedron on page 796.
Check your answer using Eulers Theorem.
SOLUTION
Counting on the diagram, the dodecahedron has 12
faces, 20 vertices, and 30 edges. Use Eulers
theorem to
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for Examples 2, 3, and 4
GUIDED PRACTICE
Describe the shape formed by the intersection of
the plane and the solid.
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for Examples 2, 3, and 4
GUIDED PRACTICE
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for Examples 2, 3, and 4
GUIDED PRACTICE
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EXAMPLE 1
Find the number of unit cubes
3- D PUZZLE
Find the volume of the puzzle piece in cubic
units.
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EXAMPLE 1
Find the number of unit cubes
SOLUTION
To find the volume, find the number of unit cubes
it contains. Separate the piece into three
rectangular boxes as follows The base is 7
units by 2 units. So, it contains 7 . 2, or 14
unit cubes. The upper left box is 2 units by 2
units. So, it contains 2 . 2, or 4 unit cubes.
The upper right box is 1 unit by 2 units. So, it
contains 1 . 2, or 2 unit cubes.
By the Volume Addition Postulate, the total
volume of the puzzle piece is 14 4 2 20
cubic units.
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EXAMPLE 2
Find volumes of prisms and cylinders
Find the volume of the solid.
a. Right trapezoidal prism
SOLUTION
V Bh 30(5) 150cm3
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EXAMPLE 2
Find volumes of prisms and cylinders
SOLUTION
b. Right cylinder
b. The area of the base is
V Bh 81p(6) 486p 1526.81 ft3
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EXAMPLE 3
Use volume of a prism
ALGEBRA
The volume of the cube is 90 cubic inches. Find
the value of x.
SOLUTION
A side length of the cube is x inches.
V x3
Formula for volume of a cube
90 in3. x3
Substitute for V.
4.48 in. x
Find the cube root.
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for Example 1,2,and 3
GUIDED PRACTICE
1. Find the volume of the puzzle piece shown
in cubic units.
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for Example 1,2,and 3
GUIDED PRACTICE
2. Find the volume of a square prism that has a
base edge length of 5 feet and a height of 12
feet.
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for Example 1,2,and 3
GUIDED PRACTICE
3. The volume of a right cylinder is 684p
cubic inches and the height is 18 inches. Find
the radius.
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EXAMPLE 4
Find the volume of an oblique cylinder
Find the volume of the oblique cylinder.
SOLUTION
Cavalieris Principle allows you to use Theorem
12.7 to find the volume of the oblique cylinder.
V p r2h
Formula for volume of a cylinder
p(42)(7)
Substitute known values.
112p
Simplify.
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EXAMPLE 4
Find the volume of an oblique cylinder
351.86
Use a calculator.
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EXAMPLE 5
Solve a real-world problem
PLANTER
The planter is made up of 13 beams. In
centimeters, suppose the dimensions of each beam
are 30 by 30 by 90. Find its volume.
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Solve a real-world problem
EXAMPLE 5
SOLUTION
The area of the base B can be found by
subtracting the area of the small rectangles from
the area of the large rectangle.
35,100 cm2
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Solve a real-world problem
EXAMPLE 5
Use the formula for the volume of a prism.
V Bh
Formula for volume of a prism
35,100(30)
Substitute.
1,053,000 cm3
Simplify.
The volume of the planter is 1,053,000 cm3, or
1.053 m3.
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for Example 4 and 5
GUIDED PRACTICE
4. Find the volume of the oblique prism shown
below.
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for Example 4 and 5
GUIDED PRACTICE
5. Find the volume of the solid shown below.
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Daily Homework Quiz

Determine whether the solid is a polyhedron. If
it is, name it.
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Daily Homework Quiz

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Daily Homework Quiz

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Daily Homework Quiz

4. Find the number of faces, vertices, and
edges of each polyhedron in Exercises 13.
5. A plane intersects a cone, but does not
intersect the base of the cone. Describe the
possible cross sections.
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Daily Homework Quiz

2. Find the volume of a right triangular prism
with height 32 in., base height 12 in., and base
length 18 in.
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Daily Homework Quiz

3. Find the volume of a right cylinder with
height 30 ft and diameter 14 ft.
4. A cylindrical beaker with diameter 10 in.
and height 12 in. is filled with water that is
then poured into a rectangular pan that is 14
in. by 9 in. by 3 in. What is the volume of
each solid? Would the water overflow the pan?
If so, what is the height of the water in the
beaker after the pan is filled?
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Daily Homework Quiz

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• Essential Questions
• When is a solid a polyhedron?
• How do you find the volume of a right prism of
  right cylinder?

• You will identify solids
• You will find volumes of prisms and cylinders

A solid is a polyhedron if it is bounded by
polygons. A polyhedron is regular if all of its
faces are congruent regular polygons. For a
polyhedron, F V E 2. The intersection of a
plane and a solid is a cross section.
If all the faces of a solid are polygons, then
the solid is a polyhedron.
For a prism, V Bh. For a cylinder, V Bh
p r 2h. Cavalieris Principle says the volume
formulas work for both right and oblique prisms
and cylinders.
The volume of a right prism or right cylinder is
the product of the height and the area of the
base.