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Three-dimensional shapes are called solids.
When a solid is formed by polygons, the solid is called a polyhedron.
The plane surfaces of a polyhedron.are called laces.
The segments joining the vertices of a polyhedron are called edges.
The two bases of a prism are congruent polygons in parallel planes.
The base of a pyramid is a polygon.
,
Tell whether the solid is a polyhedron. If so, identify the shape of the bases.
Then name the solid.
a. The solid,has a curved surface, so it is not a polyhedron.
b. The solid is formed by polygons, so it is a polyhedron. The bases are
congruent pentagons in parallel planes. This figure is a pentagonal prism.
shape of the bases. Then name the solid.
.
=
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For use with pages ~,73.~,80
Find Faces and Edges
Use the diagram at the right.
a. Name the polyhedron.
b. Count the number of faces and edges.
c. List any congruent faces and congruent edges.
a. The polyhedron is a triangular prism.
b. The polyhedron has 5 faces and 9 edges.
c. Using the markings on the diagram, you .can conclude the following.
Congruent faces:
/~ABC ~-- /kDEF
ACFD ~-- CBEF ~-- BADE
Name the polyhedron. Count the number et~ faces and edges
E
.
6. F G
H
A
Euler’s Formula relates the number of faces (F), vertices (V), and
edges (E) of a polyhedron by the equation F + V = E + 2. Use
Euler’s Formula to find the number of vertices on the hexagonal
pyramid shown.
The hexagonal pyramid has 7 faces and 12 edges.
Euler’s Formula
F+V=E+2
7+V=12+2
Substitute 7 for F and 12 for E.
7+V=14
Simplify.
Subtract 7 from each side.
V=7
Answer: The hexagonal pyramid has 7 vertices..
Use Eu~er’s Fermu~a to find the number of faces, edges, er vertices.
7. A pyramid has 9 faces and 9 vertices. How many edges does it have?
8. A prism has 8 faces and 18 edges. How many vertices does it have?
9. A polyhedron has 15 edges and 7 vertices. How many faces does it have?
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Chapter 9 Re,source Book
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A prism is a polyhedron with two congruent faces that lie in parallel planes.
The surface area of a polyhedron is the sum of the areas of its faces.
The lateral faces of a prism are the faces of the prism that are not bases.
I The lateral area of a prism is the sum of the areas of the lateral faces.
A cylinder is a solid with two congruent circular bases that li~ in parallel planes.
Surface Area of a Prism
Surface area = 2(area of base) + (perimeter of base)(height) = 2B + Ph
Surface Area of a Cylinder
Surface area = 2(area of base) + (circumference of base)(height) = 2~rr2 + 2~rrh
I
I
Find the surface area of the triangular prism.
If you visualize unfolding the prism and laying it flat,
the flat representation of the prism is called a net.
5
The surface area of a prism is equal to the area of
the net. So, to find the surface area of this triangular 7
prism, you add the areas of the triangles and rectangles 5
that make up the net.
13
13
13
1
30=
The top face and the bottom face are congruent triangles. Area: ~(12)(5)
The front face is a rectangle. Area: (12)(7)= 84
The side face is a rectangle. Area: (5)(7) = 35
The back face is a rectangle. Area: (13)(7) = 91
Add the area of all five faces.
S = 30 + 30 + 84 + 35 + 91 = 270
Answer: The surface area of the prism is 270 square units.
Draw the net of the prism in the diagram° Then find the
2.
15 cm
10 cm
12 cm
,9in.
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Find the surface area of the prism.
6m
Use the formula: Surface area = 2(area of base) + (perimeter of base)(height)
The base of the prism is a square with side length 6. So, (area of base) = 62 = 36.
The perimeter of the base is 6 + 6 + 6 + 6 - 24.
The height of the prism is 8.
Surface area = 2(36) + (24)(8) = 72 + 192 = 264
Answer: The surface area of the prism is 264 square meters.
3.
9ft
4.
5.
8 yd
24 cm
Find the surface area of the cylinder, Round your answer
to the nearest whole number.
5 yd
2 yd
2qrr2 + 2~rrh
Write the formula for surface area.
2
2~r(5) + 2rr(5)(2)
Substitute 5 for r and 2 for h.
50rr + 20¢r
Simplify.
70~r
Add.
220
Multiply.
Answer: The surface area of the cylinder is about 220 square yards.
....................................................
Find the surface area of the ~,y~inder. Round your answer to
the nearest whole number.
8.
11 m
~18m
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7. }~30 ft--------4
~15ft
8.
6in.
8i~.
Chapter 9 Resource Book
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For use wi~h pa~es ~9~-~99
1
A pyramid is a polyhedron in which the base is a polygon and the lateral faces are
triangles with a common vertex.
The height of a pyramid is the perpendicular distance between the vertex and base.
The slant height of a pyramid is the height of any of its lateral faces.
A cone has a circular base and a vertex that is not in the same plane as the base.
The height of a cone is the perpendicular distance between the vertex and the base.
The slant height of a cone is the distance between the vertex and a point on the
base edge.
SurNce Area of a lPyrarnid 1
~p
Surface area = (area of base) + -~(perimeter of base)(slant height) = B +
Surface Area of a Cone
Surface area = (area of base) + ~r(radius of base)(slant height) = ~rr2 + ~rr2
Find the slant height of the cone. Round your answer to the
nearest whole number.
12 cm
The slant height is the length of the hypotenuse of the
triangle formed by the height and the radius.
ca
(slant heighO2 = (height)2 + (radius)2
Use the Pythagorean Theorem.
= 122 + 72
Substitute 12 for height and 7 for radius.
Multiply.
= 144 + 49
Simplify.
= 193
Take the positive square root of each side.
slant, height ~ 13.89
Answer: The slant height of the cone is about 14 centimeters.
Find the slant height of the pyramid or cone° Round gout
answer to the nearest who~e number.
3,
1,
2.
3 in.
20 in.
’
10rn
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Find the Sudace Area of a PFamid
Find the surface area of the pyramid. Round your
answer to the nearest whole number.
15ft
18ft
18ft
First, find the area of the base.
B = 18 × 18 = 32z~
Then find the perimeter of the base. P = 18 + 18 + 18 + 18 = 72
Then find the slant height.
(slant height)2 = (height)2 + (½side)2 Use the Pythagorean Theorem.
= 152 + 92
Substitute. Half of 18 is 9.
= 306
Simplify.
slant height ~ 17.49
Take positive square root of each side.
Then substitute values into the formula for surface area of a pyramid.
S B+ ~P2
Write the formula for surface area of a pyramid.
~ 324 + +(72)(17.49)
~ 954 ft2
Substitute.
Simplify.
E~erc~se for Example 2
4. Find the surface area of the pyramid.
2m
Find the Surface Area of a Cone
Find the surface area of the cone with a radius of 7 centimeters and a slant
height of 12 centimeters. Round your answer to the nearest whole number.
~rr2 +
"tr (7)2 + "n’(7)(12)
418 cm2
Write the formula for surface area of a cone.
Substitute.
Simplify.
Exercise ¢#r E~ampie 3
Find the surface area of the cone. Round your answer to the
nearest whole number.
9m
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9 Resource Book
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For use with pages 58g-~07
F~n~ the v~es of pfis~s and cylinders.
~’~~~y
The wlume of a solid is the number of cubic units contained in its interior.
V~lume ~f a Prism
Volume = (~ea of base)(height) = Bh
V~lume ~f a Cylinder ,
Volume = (~ea of base)(height) = ~r~h
Find the volume of the prism.
60
5 cm
12 crn
V = Bh
Write the formula for volume of a prism.
= ¯ 5" 12 ¯ 6
Area of triangular base = -~o 5 . 12.
= 180
Simplify.
Answer: The volume of the triangular prism is 180 cubic centimeters.
Find the vo~ur~e of the prism,
12yd
2o
3.
5m
7 m
16ft
6m
lOft
12ft
Find the volume of the cylinder. Round your
answer to the nearest whole number.
3m
,25 m
Write the formula for volume of a cylinder.
= ~r(3z)(25)
Substitute 3 for r and 25 for h.
= 225~r
Simplify.
-~ 706.9
Multiply.
Answer: The volume of the cylinder is about 707 cubic meters.
V = ~rr2h
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For use with pages 580-507
Find the volume of the cyBinder. Round your answer to the
nearest who~e number.
4.
5, ~10 crn~
6 ft
6.
1 ft
14 in.
-]-
17 cm
Find the volume of the combined prisms.
4i
Find the volume of each prism, and then add the
values to find the volume of the solid.
Volume of prism A:
V = Bh
= (6.4). 10
= 240
Volume of prism B:
28 in.
¯
6 in.
9 in.
Write the formuta for volume of a prism.
Area of rectangular base is 6 ¯ 4.
Simplify.
V= Bh
Write the formula for volume of a prism.
-- (9 ¯ 4) ¯ 7
Area of rectangular base is 9 ¯ 4.
--- 252
Simplify.
Add the two values to find the volume of the entire solid.
g ,,__ 240 + 252 = 492
Answer." The volume of the combined prisms is 492 cubic inches.
5 cm
.
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Volume of a Pyramid
Volume = -](area of base)(height) = h
Volume of a Cone
1
12
Volume =.-](area of base)(height) = -]wr h
Find the volume of the pyramid.
7m
¯3m
3m
Write the formula for volume of a pyramid.
Area of square base = 32.
= ½(32)(7)
= 21 m3
Simplify.
d
9 yd
8m
15cm
.~nn
4m
11 cm
cm
Find the golume of a Cone
Find the volume of the cone. Round your
answer to the nearest whole number.
5 yd
$@L~TJ~
1 2h
V= -]~r
= ~,rr(5)2(6)
~ 157yd3
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Write the formula for volume of a cone:
Substitute 5 for r and 6 for h.
Multiply.
Chapter 9 Resource Book
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For use with pages 508-516
5. 1-- 6 rn---{
6. /~
21 cm
8m
~14 crn-{
Find the volume of the pyramid.
12 cm
You are given the slant height of the pyramid.
I0 cm
You need to find the height before you can find
10
cm
1
the volume. The height, ~ of one of the sides of
the base, and the slant height form a right triangle.
base side + (height)2 = (slant heighO2 Use the Pythagorean Theorem.
(-12.10)2+ h2= 122
25 + h2 = 144
h2 = 119
h ~ 10.9
Substitute.
Simplify.
Subtract 25 from each side.
Take positive square root of each side.
Now find the volume of the pyramid.
Write the formula for volume, of pyramid.
V= ½Bh
Area of square base = 102.
~ ½(102)(10.9)
Simplify.
--~ 363.3
363 cubic centimeters.
Answer: The volume of the pyramid is about
Find the volume of the solid. Nound ~,our answer to the
nearest who~e number.
8. ~~14cm ’
7.
6m
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For use with pages 517-523
A sphere is the set of all points in space that are the same distance from
a point, called the center of the sphere.
A geometric plane passing t ,hrough the center of a sphere divides the
sphere into two hemispheres.
Surface Area of a Sphere
Surface Area = 4~r(radius)2 = 4~rrz
Volume of a Sphere
4
3 4=3-~rr
Volume = g~r(radius)
Find the surface area of the sphere. Round your answer to the
nearest whole number.
You are given the diameter of the sphere, which is 18 feet.
You need the radius of the sphere to use the formula for
surface area. The diameter is 18, so the radius is 9.
4~r2
4qr(9)2
1018 ftz
Write the formula for surface area of a sphere.
Substitute 9 for r.
Multiply.
Exomises for E~amp~e I
/
1.
2.
20 cm
.
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For ase wi~h pages 5~7-523
Find the volume of the sphere. Round your
answer to the nearest whole number.
Write the formula for volume of a sphere.
-~’trr
~¢r(12)3
7238 in.3
Substitute 12 for r.
Multiply.
Find the volume of the sphere. Round your answer to the
nearest who~e number.
,
=
Find the volume of the ,hemisphere. Round your
answer to the nearest whole number.
A hemisphere has half the volume of a sphere.
1/4
)
Write the formula for1-] the volume of a sphere.
Substitute 8 for r.
1072 cm3
Multiply.
Nnd the v~ume o~ the hemisphere. Round y~uv answer t~
the ~earest who~e numbe~=
10.
30 ft
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