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NAME _ _ _ _ __ _ _ _ __ _ _ _ _ _ _ DATE _ _ _ __ PERIOD
Enrichment
NAME ~----------------- DATE _ _ __
PERIOD
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Reading to Learn Mathematics
Volumes of Pyramids and Cones
Pre-Activity How d o architects use geometry?
Doubling Sizes
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Consider wha t happens to surface area when t he sides of a figure are doubled.
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The sides of the large cu be are twice the size of the sides of the
small cube.
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1. How long are the edges of the large cube? 6
Read the introduction to Lesson 13-2 at the top of page 696 in your textbook.
In addition to reflecting more light, why do you think the architect of the
2. What is the surface area of the small cube? 54 in2
Transamerica Pyramid may have designed the building as a square pyramid
rather than a rectangular prism? Sample answer: The pyramid is
3. What i s the surface area of the large cube? 216 in2
more unusual and has a more dramatic appearance, so it
attracts more attention. With the sharp point at the top, it
seems to soar up into the sky.
4. The surface area of the large cube is h ow many times greater
than that of t he small cu be? 4 times
The radius of the large sphere at the right is twice the radius of
t he small sphere.
Reading the Lesson
5. What is the surface a r ea of t he small s phere? 400'1T m 2
1. In each case, two solids are described. Determine whether the first solid or the second
solid has the greater volume, or if the two solids have the same volume. (Answer by
writing first, second, or same.)
a. First solid: A rectangular prism with length x , widt hy, and height z
Second solid: A rectangular prism with length 2x, widthy, heightz second
b. First solid: a rectangular prism that has a square base with side length x and that
has heighty
Second solid: a square pyramid whose base has side length x and that has height y first
c. First solid: a right cone whose base h as radius x and that has height y
Second solid: an oblique cone whose base h as radius x and that has height y same
d. First solid: a cone whose base has radius x, and whose height is y
Second solid: a cylinder whose bases have radius x , and whose height is y second
e. First solid: a cone whose base has radius x and whose height is y
Second solid: a square pyramid whose base has side length x and whose height is y first
6. What is the surface area of the large sphere? 1600'!T m2
7. The surface area of the large sphere is how many times
greater t han the su rface area of t he small sphere? 4 times
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The sides of the large cube are twice t he size of the small cube.
9. How long are t he edges of t he large cube? 10 in.
10. W'hat is t he volume of the small cube? 125
in3
11. What is t he volu me of the large cube? 1000 in 3
12. The volu me of the large cube is how many times greater t han
that of the small cube? 8 times
multiplied by ~-
The large sphere at the right has twice the radius of t he small sphere.
c. In a square pyramid, if the side length of the base is multiplied by 1.5 and the height
is doubled, the volume will be multiplied by ~.
d. In a cone, if the r;clius of the base is tripled and the height is doubled, the volume
13. What is the volume of the small sphere? 36'1T m 3
will be multiplied by .!.!_.
e. In a cube, if the edge length is multiplied by 5, the volume will be multiplied by 125.
15. The volume of the large spher e is how many times great er
than the volume of the small sphere? 8 times
14. What is the volume of the large sphere? 28811' m3
16. It appears t hat if the dimensions of a solid a re doubled, the
volume is multiplied by -- -· 8
Helping You Remember
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3. Many students find it easier to remember mathematical formulas if they can put them
in words. Use words to describe in one sentence how to find the volume of any pyramid
or cylinder. Sample answer: Multiply the area of the base by the height and
divide by 3.
Glencoe Geometry
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Now consider how doubling the dimensions affects the volume of a cube.
be multiplied by ~b. If the radius of a cylindEl!r is tripled and the height is unchanged, the volume will be
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8. It appears that if the dimensions of a solid are dou bled, the
surface area is multiplied by _ _ _. 4
2. Supply the missing n umbers to form true statements.
a. If the length, width, and height of a rectangular box are all doubled, its volume will
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Skills Practice
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Determine whether each pair of solids are similar, congruent , or neither.
similar
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De termine whethe r each pair of solids are similar, congruent, or neither.
congruent
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Congruent and Similar Solids
Congruent and Similar Solids
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For Exercises 5-8, refer to the two
similar pris ms.
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5. Find the scale factor of the two prisms.
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6 . Find the ratio of the
surfac~
For Exercises 5-8, refer to the two similar prisms.
5. Find the scale factor of the two prisms.
6. Find the ratio of the surface areas.
7. Find the ratio of t he volumes.
areas.
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8. Suppose the surface a r ea of the larger pris m is 2560 square meters. Find the surface
ar ea of the smaller p rism. 921.6 m 2
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7. Find the ratio of the volumes.
27
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8. Suppose the volume of the larger prism is 810 cubic centimeters. Find the volume of the
smaller pl'ism.
37.5 in. x 18 in. x 15 in.
240 cm 3
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9. MINIATURES Frank Lloyd Wright designed every aspect of t he Imperial Hotel in Tokyo,
including t he chairs. The dimensions of a miniature Imperial Hotel chair are 6.25 inches x
3 inches x 2.5 inches. If the scale of the re plica is 1:6, what are the dimensions of the
original chair?
743
Glencoe Geomelry
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Glencoe Geometry
NAME
DATE
_ _ _ _ PERIOD
Reading to Learn Mathematics
Congruent and Similar Solids
NAME
DATE
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Read the introduction to Lesson 13-4 at the top of page 707 in your t extbook.
If you want t o make a miniature with a scale fact or of 1:64, how can you
use the actual object to find the measurement s you sh ould use to construct
t he miniature? Sample answer: Take linear measurements of the
actual object. Divide each measurement by 64 to find the
corresponding measurement for the miniature.
Congruent and Similar Solids
Determine wheth er each pair o f solids is simil ar, congruen t, o r neither.
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1. Determine wheth er each statement is always, sometimes , or never true.
a . Two cubes are similar. always
b. Two cones are s imilar. sometimes
c. Two cylinders in wh ich the height is twice the diameter ar e similar. always
d. Two cylinders with the same volume are congruent. sometimes
e. A prism with a square base an d a s quare pyramid are similar. never
f. Two rectangular prisms with equal su rface areas are similar. sometimes
g. Nons imilar solids h ave different volumes. sometimes
h. Two hemisph eres with the same radius are congruent. always
_ _ __ PERIOD
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Pre-Activity How are similar solids a pplie d to miniature colle c t ible s?
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2. Supply the missing rat ios.
a. If t he ratio of th e diameters of two spheres is 3: 1, then th e ratio of t heir surface areas
is
, and the ratio of their volumes is ~
9:1
5. Find the ratio of the perimeters of the bases.
b. If t he ratio of th e radii of two hemispheres is 2:5, then the ratio of their surface areas
is
4:25
, a nd the ratio of their volumes is
8: 125
c. If t wo cones are similar and the ratio of their heigh ts is
64
volumes is
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, an d t he ratio of their surface areas is ___J!_.
10:7
, and the ratio of their volumes is
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72:52 or 49:25
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d. If two cylinders are similar and the ratio of their surface areas is 100:49, then the
r atio of the radii of their bases is
7:5
6. What is the rat io of the surface areas?
t hen the r atio of their
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T h e two r ectan gular prisms shown a t the right are similar.
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7. S uppose the volume of the smaller prism is 60 in3.
Find the volume of the larger prism.
164.64 in 3
1000:343 .
Determine whether each stat e m en t is true or false. If the s t a t emen t
is false, r ewrite it so that i t is true.
Help ing You Remember
3. A good way to remember a new mathematical concept is to relate it to something you
already know. How can what y:ou know about the units used to measure lengths, areas,
and volumes help you to remember the th eorem about t he ratios of surface areas and
vol umes of similar solids? Sample answer: Lengths are measured in linear
units, surface areas In square units, and volumes in cubic units. Take the
scale factor, which is the ratio of linear measurements in the solids, and
square It to get the ratio of their surface areas or cube it to get the ratio
of their volumes.
c Glencoe/McGraw-Hill
745
Glencoe Geometry
8. If two cylinders are similar, then their volumes are equal.
False; if two cyl inders are congruent, then their volumes are equal.
9. Dou bling the height of a cylinder doubles t he volume.
true
10. 'l\vo solids are congruent if they have the same sh ape.
False; two solids are similar if they have the same shape.
c Glencoe/McG raw-Hill
746
Glencoe Geometry
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