a
Middle
School
Grade 6
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Ressource
Out Middle
High
School
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Grade 12 - Exp.
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4
3
2
1
Grade 8
/ Space, pyramid and cone
69 exercises (including 66 corrected)
a
Straight prisms and cylinders : reminders
(8 exercices)
Prisms and cylinders: enlargements and reductions
(8 exercices)
Right prisms and Pythagorean theorem
(4 exercices)
Pyramids: properties
(4 exercices)
Pyramids : volume
(2 exercices)
Pyramid and Pythagorean theorem
(6 exercices)
Pyramids and Thales' theorem
(4 exercices)
Pyramids: enlargements and reductions
(8 exercices)
Pyramids: patterns
(4 exercices)
Cones : volumes
(2 exercices)
Cones: properties, Pythagorean theorem and Thales theorem
(3 exercices)
Cones: enlargements and reductions
(3 exercices)
Cones: patterns
(2 exercices)
Complex problems and tasks
(2 exercices)
Quotient quantities and units
(3 exercices)
image PNG : /home/_math/_exercice/d9/9118/metapost/02.png
image PNG : /home/_math/_exercice/d9/9115/metapost/03.png
largeur
(
‘
Longueur
(
L
Rectangle
A
L
×
‘
p
etite
base
(
b
grande
Base
(
B
hauteur
(
h
T
rap
èze
A
B
b
×
h
2
côté
(
c
Carré
d
A
c
2
A
B
C
T
riangle
rectangle
A
AB
×
AC
2
base
(
b
hauteur
(
h
T
riangle
A
b
×
h
2
ra
y
on
(
r
Diamètre
(
D
Cercle
P
2
×
ı
×
r
A
ı
×
r
2
A
B
C
D
E
F
13
m
12
m
5
m
6
m
G
H
I
J
K
L
M
N
7
cm
4
cm
3
cm
3
cm
hauteur
(
h
Prisme
droit
V
A
B
×
h
ChingQuizz
:
2
exercises
a
v
ailable
for
Quizz
assessmen
t
:
1.
Str
aight
prisms
and
cylinders
:
r
eminders
E.9112
1
Consider
the
righ
t
prism
P
whose
base
is
a
nonagon.
Ho
w
man
y
faces
do
es
the
righ
t
prism
P
ha
v
e?
2
The
righ
t
prism
Q
has
15
edges.
What
is
the
nature
of
its
base?
Definition
:
names
of
p
olygons
:
Num
b
er
of
sides
Name
of
the
p
olygon
3
triangle
4
quadrilateral
5
p
en
tagon
6
hexagon
Num
b
er
of
sides
Name
of
the
p
olygon
7
heptagon
8
o
ctagon
9
nonagon
10
decagon
E.9058
Reminder
:
Belo
w
are
the
form
ulas
for
calculating
the
ar-
eas
of
the
follo
wing
p
olygons
:
Consider
the
t
w
o
righ
t
prisms
sho
wn
b
elo
w
in
ca
v
alier
p
er-
sp
ectiv
e
:
1
Giv
e
the
nature
of
the
base
of
eac
h
of
these
righ
t
prisms
and
determine
the
area
of
their
resp
ectiv
e
bases.
Reminder
:
Opp
osite
is
the
form
ula
for
the
v
olume
of
a
righ
t
prism,
where
A
B
represen
ts
the
area
of
its
base
and
h
repre-
sen
ts
the
measuremen
t
of
its
heigh
t.
2
Determine
the
v
olume
of
eac
h
of
these
righ
t
prisms.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/9112
sacados/9112
chapExoCorrec/9058
sacados/9058
largeur
(
‘
Longueur
(
L
Rectangle
A
L
×
‘
p
etite
base
(
b
grande
Base
(
B
hauteur
(
h
T
rap
èze
A
B
b
×
h
2
côté
(
c
Carré
d
A
c
2
A
B
C
T
riangle
rectangle
A
AB
×
AC
2
base
(
b
hauteur
(
h
T
riangle
A
b
×
h
2
ra
y
on
(
r
Diamètre
(
D
Cercle
P
2
×
ı
×
r
A
ı
×
r
2
A
B
C
D
E
F
13
m
12
m
5
m
6
m
G
H
I
J
K
L
M
N
7
cm
4
cm
3
cm
3
cm
hauteur
(
h
Prisme
droit
V
A
B
×
h
(
d
hauteur
(
h
ra
y
on
(
r
Cylindre
S
latérale
2
×
ı
×
r
×
h
V
ı
×
r
2
×
h
15
cm
3
cm
3
cm
4
cm
5
cm
8
cm
12
cm
15
cm
20
cm
A
B
C
D
E
F
E.6449
In
the
table
b
elo
w,
for
eac
h
ro
w,
re-
triev
e
the
v
olume
v
alue
on
the
left
and
con
v
ert
it
using
the
unit
sho
wn
on
the
righ
t
:
k
m
3
hm
3
dam
3
m
3
dm
3
cm
3
mm
3
312
m
3
.
.
.
dm
3
0
;
32
dm
3
.
.
.
m
3
350
mm
3
.
.
.
m
3
2
‘
.
.
.
m
3
33
c‘
.
.
.
cm
3
25
k
m
3
.
.
.
m
3
R
e
c
al
l
the
e
quality
:
1
‘
=
1
dm
3
E.9059
1
a
A
righ
t
prism
has
a
triangular
base.
Ho
w
man
y
faces
do
es
this
solid
ha
v
e?
b
A
righ
t
prism
has
a
hexagonal
base.
Ho
w
man
y
edges
do
es
it
ha
v
e?
2
a
A
righ
t
prism
has
12
edges.
What
is
the
nature
of
its
base?
b
A
righ
t
prism
has
7
faces.
What
is
the
nature
of
its
base?
E.9064
A
tiered
cak
e
consists
of
3
cylindri-
cal
cak
es
stac
k
ed
on
top
of
eac
h
other,
all
cen
tered
on
the
axis
(
d
)
,
as
sho
wn
in
the
figure
b
elo
w
:
The
three
tiers
of
the
cak
e
are
all
8
cm
high,
and
the
resp
ectiv
e
diameters
of
these
cylinders
are
30
cm
,
20
cm
,
and
10
cm
.
Determine
the
v
olume
of
this
cak
e.
Round
the
v
olume
to
the
nearest
cubic
cen
timeter.
Reminder
:
The
lateral
surface
area
and
v
olume
of
a
cylinder
are
giv
en
opp
osite,
as
a
function
of
the
radius
of
its
base
and
its
heigh
t.
Hin
t
:
w
e
will
use
:
ı
≈
3
;
1416
w
e
will
round
the
in
termediate
results
to
the
nearest
cubic
millimeter.
E.9110
Belo
w
is
the
pattern
for
a
cylinder
:
Determine
the
heigh
t
of
the
cylinder
and
the
radius
of
its
base.
If
necessary
,
round
the
v
alues
to
the
nearest
millimeter.
Note
:
w
e
will
use
:
ı
≈
3
;
1416
E.9659
W
e
consider
the
righ
t
prism
b
elo
w
:
Determine
the
v
olume
of
this
solid.
E.9660
Consider
the
righ
t
prism
b
elo
w
whose
base
is
a
righ
t
triangle
in
A
:
Determine
the
v
olume
of
this
solid.
2.
Prisms
and
cylinders
:
enlar
gements
and
r
e
ductions
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6449
sacados/6449
chapExoCorrec/9059
sacados/9059
chapExoCorrec/9064
sacados/9064
(
d
hauteur
(
h
ra
y
on
(
r
Cylindre
S
latérale
2
×
ı
×
r
×
h
V
ı
×
r
2
×
h
chapExoCorrec/9110
sacados/9110
15
cm
3
cm
sacados/9659
3
cm
4
cm
5
cm
8
cm
sacados/9660
12
cm
15
cm
20
cm
A
B
C
D
E
F
10
cm
4
cm
2
cm
A
B
C
D
E
F
G
H
M
N
P
Q
R
S
T
A
B
C
D
E
F
G
H
A
B
E
F
B
A
A
E.9124
Definition
:
Let
k
b
e
a
strictly
p
ositiv
e
n
um
b
er.
F
or
k
<
1
and
when
all
dimensions
of
a
plane
figure
or
solid
are
m
ultiplied
b
y
k
,
w
e
sa
y
that
w
e
ha
v
e
p
erformed
a
reduction
b
y
a
factor
of
k
.
F
or
k
>
1
and
when
all
the
dimensions
of
a
plane
figure
or
a
solid
are
m
ultiplied
b
y
k
,
w
e
sa
y
that
w
e
ha
v
e
p
erformed
an
enlargemen
t
b
y
a
co
efficien
t
of
k
.
Consider
the
righ
t-angled
parallelepip
ed
AB
C
D
E
F
GH
sho
wn
b
elo
w,
with
dimensions
:
AB
=
10
cm
;
B
C
=
4
cm
;
C
G
=
2
cm
W
e
construct
the
righ
t-angled
blo
c
k
M
N
P
D
QR
S
T
obtained
from
the
righ
t-angled
blo
c
k
AB
C
D
E
F
GH
b
y
reducing
its
dimensions
b
y
a
factor
of
1
2
.
1
Noting
A
as
the
area
of
face
AB
F
E
and
A
as
the
area
of
face
M
N
R
Q
,
determine
the
v
alue
of
the
quotien
t
:
A
A
2
Let
V
b
e
the
v
olume
of
the
rectangular
prism
AB
C
D
E
F
GH
and
V
b
e
the
v
olume
of
the
rectangu-
lar
prism
M
N
P
D
QR
S
T
.
Determine
the
v
alue
of
the
quotien
t
:
V
V
E.980
Prop
osition
:
If
all
dimensions
of
a
plane
figure
is
reduced
or
enlarged
b
y
a
co
efficien
t
k
then
its
area
has
b
een
m
ultiplied
b
y
k
2
.
If
all
dimensions
of
a
solid
is
reduced
or
enlarged
b
y
a
co
efficien
t
k
then
its
v
olume
has
b
een
m
ultiplied
b
y
k
3
.
Cop
y
the
table
b
elo
wb
elo
w
on
y
our
sheet
of
pap
er
and
com-
plete
it
using
the
square
ro
ot
k
eys
√
x
et
racines
n
ième
x
√
y
k
k
2
k
3
5
16
729
E.9118
The
Aztec
calendar
preserv
ed
in
Mexico
Cit
y’s
National
Museum
of
An
trop
ology
has
a
diameter
of
3.60
meters
and
w
eighs
25
tons.
What
w
ould
b
e
the
w
eigh
t
of
a
replica
20
cen
timetres
in
diameter?
(round
to
the
nearest
decigram)
E.9115
In
this
pan,
y
ou
can
prepare
a
paella
for
three
p
eople.
T
aking
a
pan
four
times
the
size,
ho
w
man
y
p
eople
can
w
e
in
vite?
E.9116
The
radius
of
the
large
cylinder
has
b
een
halv
ed.
By
ho
w
m
uc
h
has
its
v
ol-
ume
b
een
reduced?
E.9119
In
the
figure
opp
o-
site,
the
parallelepip
ed
A
B
C
D
E
F
G
H
w
as
ob-
tained
from
AB
C
D
E
F
GH
b
y
reducing
its
length
and
width
b
y
2.
By
ho
w
m
uc
h
has
the
v
ol-
ume
of
the
parallelepid
de
A
B
C
D
E
F
G
H
b
een
re-
duced.
E.4210
The
v
olume
of
the
parallelepip
ed
has
b
een
m
ultiplied
b
y
27.
By
ho
w
m
uc
h
ha
v
e
its
dimensions
b
een
en-
larged?
E.9122
The
v
olume
of
the
small
paral-
lelepip
ed
is
14
dm
3
.
The
ratio
B
A
B
A
of
the
heigh
ts
of
the
t
w
o
parallelepip
eds
is
1
=
3
.
Calculate
the
v
olume
of
the
large
par-
allelepip
ed.
3.
R
ight
prisms
and
Pythagor
e
an
the
or
em
h
ttps:/
/c
hingmath.fr
chapExoCorrec/9124
sacados/9124
10
cm
4
cm
2
cm
A
B
C
D
E
F
G
H
M
N
P
Q
R
S
T
chapExoCorrec/980
sacados/980
chapExoCorrec/9118
sacados/9118
chapExoCorrec/9115
sacados/9115
chapExoCorrec/9116
sacados/9116
chapExoCorrec/9119
sacados/9119
A
B
C
D
E
F
G
H
A
B
E
F
chapExoCorrec/4210
sacados/4210
chapExoCorrec/9122
sacados/9122
B
A
A
A
B
C
D
E
F
Baie
vitrée
8
m
4
;
2
m
0
;
5
m
0
;
79
m
A
B
C
D
0
;
79
m
4
;
2
m
0
;
5
m
arrière
droite
face avant
gauche
A
B
C
D
E
F
A
B
C
D
E
F
G
H
7
cm
24
cm
15
cm
10
cm
E.6450
Consider
the
righ
t
prism
AB
C
D
E
F
sho
wn
b
elo
w
:
The
follo
wing
measuremen
ts
are
giv
en
:
AB
=
6.5
cm
;
AC
=
1.6
cm
;
B
C
=
6.3
cm
;
AD
=
3
cm
1
Sho
w
that
the
triangle
AB
C
is
a
righ
t
triangle.
2
Determine
the
v
olume
of
the
righ
t
prism
AB
C
D
E
F
.
E.9068
Mrs
Martin
w
ould
lik
e
to
create
a
concrete
terrace
in
fron
t
of
her
ba
y
windo
w.
She
pro
duces
the
dra
wing
b
elo
w,
indicating
the
measure-
men
ts.
T
o
facilitate
rain
w
ater
run-off,
the
terrace
flo
or
m
ust
b
e
slop
ed.
The
terrace
is
shap
ed
lik
e
a
righ
t
prism
with
a
righ
t-angled
trap
ezoid
at
its
base.
1
Determine
the
v
olume
of
concrete
required
for
the
terrace
design.
2
Belo
w
is
a
profile
view
of
the
terrace.
a
Determine
the
length
of
the
segemen
t
[
C
D
]
.
b
Deduce
the
surface
area
of
the
terrace.
E.6452
W
e
stac
k
ed
and
glued
6
cub
es
with
edges
of
length
4
cm
and
a
righ
t
prism
to
obtain
the
solid
sho
wn
b
elo
w.
The
heigh
t
of
the
prism
is
equal
to
half
the
edge
length
of
the
cub
es.
1
Dra
w
a
full-scale
view
of
the
rear
of
the
solid.
2
Calculate
the
v
olume
in
cm
3
of
the
solid.
3
Study
of
the
righ
t
prism.
a
This
prism
is
called
AB
C
D
E
F
,
as
sho
wn
in
the
figure
b
elo
w
:
What
is
the
nature
of
the
base
of
this
righ
t
prism?
Justify
y
our
answ
er.
b
V
erify
b
y
calculation
that
the
length
is
AC
=
32
cm
.
c
Deduce
the
exact
v
alue
of
the
area
of
face
AC
F
D
.
Round
to
the
nearest
mm
2
.
E.9065
Consider
the
righ
t
prism
AB
C
D
E
D
F
GH
whose
base
AB
C
D
is
an
y
quadrilateral,
but
has
t
w
o
righ
t
angles
at
v
ertices
B
and
D
.
1
Determine
the
area
of
the
base
of
this
righ
t
prism.
2
Giv
e
the
v
olume
of
this
righ
t
prism.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6450
sacados/6450
A
B
C
D
E
F
chapExoCorrec/9068
sacados/9068
Baie
vitrée
8
m
4
;
2
m
0
;
5
m
0
;
79
m
A
B
C
D
0
;
79
m
4
;
2
m
0
;
5
m
chapExoCorrec/6452
sacados/6452
arrière
droite
face avant
gauche
A
B
C
D
E
F
chapExoCorrec/9065
sacados/9065
A
B
C
D
E
F
G
H
7
cm
24
cm
15
cm
10
cm
P
y
r
a
m
i
d
e
A
B
C
D
S
T
E
F
G
H
I
J
A
B
C
D
E
F
S
G
H
I
J
K
L
S
A
B
C
D
E
F
G
H
I
J
K
O
4.
Pyr
amids
:
pr
op
erties
E.4958
Definition
:
The
p
yramid
is
a
solid
with
a
p
olygonal
base
and
a
p
oin
t
called
v
ertex
(also
c
al
le
d
ap
ex)
whic
h
is
connected
to
all
the
v
er-
tices
of
the
base.
Prop
osition
:
If
the
base
of
a
p
yramid
has
n
v
ertices,
the
p
yramid
con
tains
2
n
edges
and
n
+
1
faces.
Consider
the
t
w
o
p
yramids
b
elo
w
:
1
Consider
the
p
yramid
AB
C
D
S
:
a
What
is
the
nature
of
the
base
of
this
p
yramid?
b
Ho
w
man
y
edges
do
es
this
p
yramid
ha
v
e?
c
Ho
w
man
y
faces
do
es
this
p
yramid
ha
v
e?
2
Consider
the
p
yramid
E
F
GH
I
J
T
:
a
What
is
the
nature
of
the
base
of
this
p
yramid?
b
Ho
w
man
y
edges
do
es
this
p
yramid
ha
v
e?
c
Ho
w
man
y
faces
do
es
this
p
yramid
ha
v
e?
E.9111
1
Consider
the
p
yramid
P
whose
base
is
a
heptagon.
Ho
w
man
y
faces
do
es
the
p
yramid
P
ha
v
e?
2
The
p
yramid
Q
has
18
edges.
What
is
the
nature
of
its
base?
Definition
:
names
of
p
olygons
:
Num
b
er
of
sides
Name
of
the
p
olygon
3
triangle
4
quadrilateral
5
p
en
tagon
6
hexagon
Num
b
er
of
sides
Name
of
the
p
olygon
7
heptagon
8
o
ctagon
9
nonagon
10
decagon
E.4952
Consider
the
t
w
o
p
yramids
AB
C
D
E
F
D
and
GH
I
J
K
LS
with
hexagonal
bases
sho
wn
b
elo
w.
The
first
p
yramid
is
random
while
the
second
is
a
regular
p
yramid.
1
Can
the
fo
ot
O
of
the
heigh
t
of
the
p
yramid
AB
C
D
E
F
S
from
S
b
e
traced
precisely
.
2
Place
the
p
oin
t
O
represen
ting
the
fo
ot
of
the
p
yramid
heigh
t
GH
I
J
K
LS
originating
from
the
v
ertex
S
.
Jus-
tify
y
our
approac
h.
Definition
:
A
p
yramid
is
said
to
b
e
regular
if
its
base
is
a
regular
p
olygon
(e
quilater
al
triangle,
squar
e,
r
e
gular
hep-
tagon.
.
.
)
and
if
the
fo
ot
of
the
heigh
t
is
the
cen
ter
of
its
base.
E.4954
Consider
the
righ
t-hand
pa
ving
stone
AB
C
D
E
F
GH
sho
wn
b
elo
w
où
:
K
is
the
middle
of
the
segmen
t
[
E
H
]
;
I
is
the
cen
ter
of
the
face
AB
C
D
;
J
is
the
cen
ter
of
the
face
E
F
GH
;
O
is
the
cen
ter
of
the
cub
oid.
Giv
e
the
natures
of
the
solids
b
elo
w
and
state
their
base
and
heigh
t
:
a
AD
C
E
H
G
b
AD
C
E
c
D
I
C
O
d
E
K
GF
5.
Pyr
amids
:
volume
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4958
sacados/4958
P
y
r
a
m
i
d
e
A
B
C
D
S
T
E
F
G
H
I
J
chapExoCorrec/9111
sacados/9111
chapExoCorrec/4952
sacados/4952
A
B
C
D
E
F
S
G
H
I
J
K
L
S
chapExoCorrec/4954
sacados/4954
A
B
C
D
E
F
G
H
I
J
K
O
P
y
r
a
m
i
d
e
A
B
C
D
E
F
G
H
O
S
12
m
8
m
4
m
3
m
A
B
C
D
H
S
230
m
146
m
A
B
C
S
15
cm
25
cm
21
cm
A
B
C
D
H
S
24
m
18
m
25
m
E.4902
Prop
osition
:
Let
P
b
e
a
p
yramid
of
heigh
t
h
and
whose
base
has
area
A
.
The
v
olume
V
of
the
p
yramid
has
measure
:
V
=
1
3
×A×
h
A
house
is
built
b
y
sup
erimp
osing
a
righ
t
blo
c
k
AB
C
D
E
F
GH
and
a
p
yramid
E
F
GH
S
with
v
ertex
S
.
The
represen
tation
b
elo
w
sp
ecifies
some
measuremen
ts
:
Determine
the
total
v
olume
of
this
house.
E.9066
The
p
yramid
of
Cheops,
lo
cated
in
Egypt,
is
a
square-based
p
yramid
whose
base
sides
measure
230
m
and
the
heigh
t,
at
its
construction,
measured
146
m
.
Here
is
a
represen
tation
of
this
p
yramid
:
1
Determine
the
v
olume
of
the
p
yramid
of
Cheops,
rounded
to
the
nearest
cubic
meter.
2
Assuming
that
all
the
stones
in
the
p
yramid
are
iden-
tical
and
that
eac
h
has
a
v
olume
of
0.95
m
3
and
that
eac
h
w
eighs
2.1
tons.
Determine,
in
kilograms,
the
total
w
eigh
t
of
the
p
yramid
of
Cheops.
6.
Pyr
amid
and
Pythagor
e
an
the
or
em
E.9130
In
space,
consider
the
p
yramid
AB
C
S
whose
base
AB
C
is
a
righ
t
triangle
and
whose
v
er-
tex
is
p
oin
t
S
.
Also,
the
AB
S
face
is
a
righ
t
triangle
in
A
and
the
C
AS
face
is
a
righ
t
triangle
in
A
.
The
follo
wing
dimensions
are
giv
en
:
AB
=
15
cm
;
AC
=
21
cm
;
B
S
=
25
cm
1
Establish
that
:
AS
=
20
cm
2
Determine
the
v
olume
of
the
p
yramid.
E.5675
Consider
the
p
yramid
AB
C
D
S
sho
wn
b
elo
w
o
where
the
base
AB
C
D
is
a
rectangle
and
H
is
the
fo
ot
of
the
heigh
t
from
the
v
ertex
S
and
the
middle
of
the
segmen
t
[
AC
]
:
1
Sho
w
that
the
segmen
t
[
AH
]
has
length
15
m
.
2
a
Determine
the
length
of
the
heigh
t
[
S
H
]
.
b
Determine
the
v
olume
of
the
p
yramid
S
AB
C
D
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4902
sacados/4902
P
y
r
a
m
i
d
e
A
B
C
D
E
F
G
H
O
S
12
m
8
m
4
m
3
m
chapExoCorrec/9066
sacados/9066
A
B
C
D
H
S
230
m
146
m
chapExoCorrec/9130
sacados/9130
A
B
C
S
15
cm
25
cm
21
cm
chapExoCorrec/5675
sacados/5675
A
B
C
D
H
S
24
m
18
m
25
m
A
B
C
S
24
cm
37
;
3
cm
34
;
8
cm
A
B
C
D
E
F
H
G
M
N
E
H
G
F
A
D
C
B
I
J
K
A
B
C
D
E.9131
In
space,
consider
the
p
yramid
AB
C
S
whose
base
AB
C
is
a
righ
t
triangle
and
whose
v
er-
tex
is
p
oin
t
S
.
Also,
the
AB
S
face
is
a
righ
t
triangle
in
A
and
the
C
AS
face
is
a
righ
t
triangle
in
A
.
The
follo
wing
dimensions
are
giv
en
:
AB
=
24
cm
;
B
C
=
34.8
cm
;
C
S
=
37.3
cm
Determine
the
v
olume
of
the
p
yramid.
E.981
AB
C
D
E
F
GH
is
a
righ
t-angled
parallelepip
ed.
M
is
a
p
oin
t
on
the
segmen
t
[
F
G
]
and
N
b
e-
longs
to
the
segmen
t
[
E
F
]
The
follo
wing
measuremen
ts
are
giv
en
:
F
E
=
12
cm
;
F
G
=
9
cm
;
F
B
=
3
cm
F
N
=
4
cm
;
F
M
=
3
cm
Here
is
a
represen
tation
of
this
configuration
:
1
Calculate
length
M
N
2
Sho
w
that
the
area
of
the
triangle
F
N
M
is
equal
to
6
cm
2
.
3
Calculate
the
v
olume
of
the
p
yramid
(
P
)
with
v
ertex
B
and
base
the
triangle
F
N
M
.
4
Consider
the
solid
AB
C
D
E
N
M
GH
obtained
b
y
remo
v-
ing
the
p
yramid
(
P
)
from
the
righ
t-angled
parallelepip
ed.
a
Ho
w
man
y
faces
do
es
this
solid
ha
v
e?
b
Calculate
its
v
olume
E.3275
Reminder
:
v
olume
V
of
a
p
yramid
:
V
=
(aire
de
la
base)
×
hauteur
3
AB
C
D
E
F
GH
is
a
cub
e
with
edge
length
AB
=
12
cm
.
I
is
the
midp
oin
t
of
segmen
t
[
AB
]
;
J
is
the
midp
oin
t
of
segmen
t
[
AE
]
;
K
is
the
midp
oin
t
of
segmen
t
[
AD
]
.
1
Calculate
the
area
of
triangle
AI
K
.
2
Calculate
the
v
olume
of
p
yramid
AI
K
J
with
base
AK
I
.
3
What
fraction
of
the
v
olume
of
the
cub
e
do
es
the
v
ol-
ume
of
p
yramid
AI
K
J
represen
t?
W
rite
the
result
as
a
fraction
with
n
umerator
1
.
4
Dra
w
a
template
of
p
yramid
AI
K
J
.
E.4951
Consider
the
p
yramid
AB
C
D
with
a
triangular
base
:
AB
=
B
C
=
B
D
;
AC
=
AD
=
C
D
=
5
cm
In
addition,
the
AB
D
,
AB
C
,
and
B
C
D
faces
are
righ
t-angled
triangles
at
B
.
1
In
the
triangle
AB
C
,
determine
the
measure
of
segmen
t
[
AB
]
rounded
to
the
nearest
millimeter.
2
Determine
the
v
olume
of
the
p
yramid
AB
C
D
rounded
to
the
nearest
cm
3
.
7.
Pyr
amids
and
Thales’
the
or
em
h
ttps:/
/c
hingmath.fr
chapExoCorrec/9131
sacados/9131
A
B
C
S
24
cm
37
;
3
cm
34
;
8
cm
chapExoCorrec/981
sacados/981
Groupe Sud - Juin 2003 - 7 points
A
B
C
D
E
F
H
G
M
N
chapExoCorrec/3275
sacados/3275
Brevet juin 2010 - 6 points
E
H
G
F
A
D
C
B
I
J
K
chapExoCorrec/4951
sacados/4951
A
B
C
D
A
B
C
D
M
N
P
Q
H
S
H
A
B
C
D
E
F
S
H
A
B
C
D
M
N
P
Q
H
S
H
A
B
C
D
H
S
M
N
P
Q
H
S
E.9125
Consider
the
p
yramid
AB
C
D
S
whose
base
AB
C
D
is
a
square
suc
h
that
:
AB
=
9
cm
;
S
H
=
12.75
cm
;
S
B
=
14.25
cm
W
e
construct
the
p
yramid
M
N
P
QS
whose
base
M
N
P
Q
is
a
square
parallel
to
the
square
AB
C
D
and
suc
h
that
:
S
N
=
9.5
cm
Determine
the
measure
of
heigh
t
[
S
H
]
of
the
p
yramid
M
N
P
QS
.
E.9106
W
e
w
an
t
to
build
a
teep
ee
in
the
shap
e
of
a
p
yramid
with
a
rect-
angular
base
AB
C
D
cen
tered
at
H
and
heigh
t
[
S
H
]
(se
e
the
diagr
am
opp
osite)
.
The
teep
ee
will
ha
v
e
the
follo
w-
ing
dimensions
:
AD
=
1
;
60
m
;
C
D
=
1
;
20
m
S
H
=
2
;
40
m
;
S
F
=
1
;
95
m
S
D
=
2
;
60
m
1
Calculate
the
v
olume
V
of
this
p
yramid
in
m
3
.
Reminders
:
V
=
1
3
×
B
×
h
is
the
area
of
the
p
yramid,
where
h
is
the
heigh
t
and
B
is
the
area
of
the
base.
The
frame
of
the
teep
ee,
consisting
of
the
rectangular
frame
AB
C
D
and
the
four
side
edges
extending
from
S
,
is
made
of
bam
b
o
o
stic
ks.
A
ro
d
[
E
F
]
is
added
to
the
frame
as
sho
wn
in
the
dra
wing
so
that
(
E
F
)
=
=
(
AD
)
and
S
F
=
1
;
95
m
.
2
Calculate
E
F
.
E.9127
Consider
the
p
yramid
AB
C
D
S
whose
base
AB
C
D
is
a
square
suc
h
that
:
AB
=
9
cm
;
S
H
=
12.75
cm
;
S
B
=
14.25
cm
W
e
construct
the
p
yramid
M
N
P
QS
whose
base
M
N
P
Q
is
a
square
parallel
to
the
square
AB
C
D
and
suc
h
that
:
S
H
=
8.5
cm
1
Determine
the
v
olume
V
of
the
p
yramid
AB
C
D
.
2
a
Establish
that
:
S
N
=
9.5
cm
b
Establish
that
:
M
N
=
6
cm
c
Determine
the
v
olume
V
of
the
p
yramid
M
N
P
QS
.
3
Establish
the
follo
wing
equalities
:
S
N
S
B
=
M
N
AB
=
2
3
;
V
V
=
2
3
3
Hin
t
:
the
calculator
can
b
e
used
to
reduce
fractions.
E.9128
Consider
the
p
yramid
AB
C
D
S
whose
base
AB
C
D
is
a
rectangle
and
whose
v
ertex
is
S
.
The
follo
wing
dimensions
are
giv
en
:
AB
=
24
m
;
B
C
=
18
m
;
S
B
=
39
m
1
a
Sho
w
that
the
segmen
t
[
B
H
]
measures
15
m
.
b
Deduce
that
:
S
H
=
36
m
c
Giv
e
the
v
olume
of
the
p
yramid
AB
C
D
S
.
The
p
oin
t
N
b
elongs
to
the
segmen
t
[
S
B
]
suc
h
that
:
S
N
=
26
m
W
e
create
a
section
of
the
p
yramid
from
the
p
oin
t
N
b
y
a
plane
parallel
to
the
base
of
the
p
yramid.
W
e
denote
H
the
p
oin
t
of
in
tersection
of
the
cutting
plane
with
the
segmen
t
[
S
H
]
.
The
solid
M
N
P
QS
is
a
p
yramid
with
a
rectangular
base
M
N
P
Q
and
a
v
ertex
S
,
and
the
p
oin
t
H
is
the
fo
ot
of
the
heigh
t
from
S
.
2
Establish
that
:
M
N
=
16
m
;
S
H
=
24
m
3
Determine
the
measure
of
segmen
t
[
S
H
]
.
4
It
is
assumed
that
N
P
=
12
m
.
Deduce
the
v
olume,
to
the
nearest
cubic
meter,
of
the
p
yramid
M
N
P
QS
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/9125
sacados/9125
A
B
C
D
M
N
P
Q
H
S
H
chapExoCorrec/9106
sacados/9106
A
B
C
D
E
F
S
H
chapExoCorrec/9127
sacados/9127
A
B
C
D
M
N
P
Q
H
S
H
chapExoCorrec/9128
sacados/9128
A
B
C
D
H
S
M
N
P
Q
H
S
A
B
C
D
H
S
M
N
P
Q
H
S
A
B
C
D
M
N
P
Q
H
S
H
O
A
B
C
D
E
I
J
K
L
H
H
H
H
8.
Pyr
amids
:
enlar
gements
and
r
e
ductions
E.9104
Consider
the
p
yramid
AB
C
D
S
whose
base
AB
C
D
is
a
rectangle
and
whose
v
ertex
is
S
.
The
follo
wing
dimensions
are
giv
en
:
AB
=
8
m
;
B
C
=
6
m
;
S
H
=
12
m
;
S
H
=
9
m
1
Determine
the
v
olume
of
the
p
yramid
AB
C
D
S
.
2
a
Justify
that
the
reduction
to
obtain
the
p
yramid
M
N
P
QS
has
a
reduction
co
efficien
t
of
3
4
.
b
Deduce
the
v
olume
of
the
p
yramid
M
N
P
QS
.
E.9109
Definition
:
A
p
yramid
is
said
to
b
e
regular
if
its
base
is
a
regular
p
olygon
(e
quilater
al
triangle,
squar
e,
r
e
gular
p
en-
tagon.
.
.
)
and
if
the
fo
ot
of
the
heigh
t
from
the
ap
ex
of
the
p
yramid
is
the
cen
ter
of
its
base.
W
e
consider
the
p
yramid
AB
C
D
S
to
b
e
regular,
whose
base
AB
C
D
is
a
square
and
the
fo
ot
of
the
heigh
t
from
S
is
the
p
oin
t
H
.
W
e
ha
v
e
the
measuremen
ts
:
AB
=
9
cm
;
S
H
=
12.75
cm
;
S
B
=
14.25
cm
W
e
construct
the
p
yramid
M
N
P
QS
whose
base
M
N
P
Q
is
a
square
parallel
to
the
square
AB
C
D
.
W
e
denote
H
as
the
fo
ot
of
the
heigh
t
of
this
p
yramid
from
the
v
ertex
S
.
W
e
ha
v
e
the
measuremen
t
:
S
H
=
8.5
cm
1
Determine
the
v
olume
V
of
the
p
yramid
AB
C
D
S
.
2
Determine
the
length
of
segmen
t
[
S
N
]
.
Hin
t
:
w
e
assume
that
lines
(
H
B
)
and
(
H
N
)
are
parallel.
3
a
Determine
the
reduction
co
efficien
t
applied
to
the
dimensions
of
the
p
yramid
AB
C
D
S
to
obtain
the
p
yra-
mid
M
N
P
QS
.
b
Determine
the
v
olume
V
of
the
p
yramid
M
N
P
QS
.
E.4953
Consider
the
AB
C
D
E
p
yramid
with
a
rectangular
base
sho
wn
b
elo
w
and
suc
h
that
[
O
E
]
is
the
heigh
t
of
the
p
yramid
and
where
O
is
the
cen
ter
of
the
base
:
The
p
oin
ts
I
,
J
,
K
,
and
L
represen
t
the
resp
ectiv
e
middles
of
the
segmen
ts
[
E
A
]
,
[
E
B
]
,
[
E
C
]
,
and
[
E
D
]
.
W
e
assume
that
:
The
solid
I
J
K
LE
is
a
p
yramid
with
rectangular
base
I
J
K
L
and
v
ertex
S
;
bases
AB
C
D
and
I
J
K
L
are
parallel
and
w
e
also
ha
v
e
the
relations
:
(
I
J
)
=
=
(
AB
)
;
(
J
K
)
=
=
(
B
C
)
AB
=
16
cm
;
B
C
=
12
cm
;
E
B
=
26
cm
1
a
Justify
that
the
segmen
t
[
K
J
]
measures
6
cm
b
W
e
amdet
that
the
base
I
J
K
L
of
area
A
is
a
reduc-
tion
of
the
base
AB
C
D
of
area
A
.
Giv
e
the
reduction
co
efficien
t
of
the
area
A
to
obtain
the
area
A
2
It
is
assumed
that
O
E
=
24
cm
.
Determine
the
v
olumes
V
of
the
p
yramid
AB
C
D
E
and
V
of
the
p
yramid
I
J
K
LE
.
E.9120
The
v
olume
of
the
Great
Pyramid
is
576
k
m
3
.
The
v
olume
of
the
small
p
yramid
is
9
k
m
3
.
Establish
equalit
y
:
S
H
S
H
=
1
4
E.984
The
heigh
t
of
the
Great
Pyramid
measures
32
cm
.
The
heigh
t
of
the
small
p
yramid
measures
8
cm
.
The
v
olume
of
the
small
p
yramid
is
13
cm
3
.
Calculate
the
v
olume
of
the
large
p
yra-
mid.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/9104
sacados/9104
A
B
C
D
H
S
M
N
P
Q
H
S
chapExoCorrec/9109
sacados/9109
A
B
C
D
M
N
P
Q
H
S
H
chapExoCorrec/4953
sacados/4953
O
A
B
C
D
E
I
J
K
L
chapExoCorrec/9120
sacados/9120
H
H
chapExoCorrec/984
sacados/984
H
H
A
B
C
D
E
F
G
H
S
A
B
C
D
E
F
G
H
I
O
I
S
A
B
C
D
A
B
C
D
E.982
The
house
mo
del
sho
wn
b
elo
w
consists
of
:
of
a
righ
t
blo
c
k
of
dimensions
:
AB
=
30
cm
;
AE
=
20
cm
;
AD
=
5
cm
topp
ed
b
y
a
p
yramid
of
heigh
t
6
cm
.
1
Calculate
the
v
olume
V
1
of
this
mo
del.
2
Kno
wing
that
this
mo
del
is
a
co
efficien
t
reduction
1
=
50
of
the
real
house,
deduce
from
the
first
question
the
v
olume
V
2
in
m
3
of
the
house.
E.986
The
dimension
reduction
co
effi-
cien
t
is
1
2
.
Kno
wing
that
the
area
of
the
base
of
the
large
p
yramid
is
24
m
2
,
cal-
culate
the
area
of
the
base
of
the
small
p
yramid.
Kno
wing
that
the
v
olume
of
the
small
p
yramid
is
56
m
3
,
calculate
the
v
olume
of
the
large
p
yramid.
E.9117
The
S
A
B
C
D
p
yramid
w
as
ob-
tained
b
y
sectioning
the
S
AB
C
D
p
yramid
using
a
plane
parallel
to
the
latter’s
base.
Kno
wing
that
the
v
olume
has
b
een
reduced
b
y
64,
determine
the
co
ef-
ficien
t
of
reduction
of
the
area
of
their
common
bases.
9.
Pyr
amids
:
p
atterns
E.4959
Cut
out
the
pattern
b
elo
w
of
a
tri-
angular
based
p
yramid
and
reconstruct
the
solid.
E.4960
Cut
out
the
pattern
b
elo
w
of
a
tri-
angular
based
p
yramid
and
reconstruct
the
solid.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/982
sacados/982
Brevet - 3 points
A
B
C
D
E
F
G
H
S
chapExoCorrec/986
sacados/986
A
B
C
D
E
F
G
H
I
O
I
chapExoCorrec/9117
sacados/9117
S
A
B
C
D
A
B
C
D
chapExoCorrec/4959
sacados/4959
chapExoCorrec/4960
sacados/4960
A
B
C
D
E
F
G
H
I
J
K
Schéma
1
Schéma
2
Schéma
3
Schéma
4
A
B
C
D
E
F
G
H
r
Cône
hauteur
(
h
E.7979
The
p
yramid
F
I
J
K
is
cut
out
of
the
cub
e
AB
C
D
E
F
GH
as
sho
wn
in
the
adjacen
t
dra
wing.
The
segmen
t
[
AB
]
measures
6
cm
.
The
p
oin
ts
I
,
J
and
K
are
the
resp
ectiv
e
middles
of
the
edges
[
F
E
]
,
[
F
B
]
and
[
F
G
]
.
1
Dra
w
the
triangle
I
F
K
full
size.
2
One
of
the
four
diagrams
b
elo
w
corresp
onds
to
the
p
yra-
mid
pattern
F
I
J
K
.
Indicate
its
n
um
b
er
on
the
cop
y
.
No
justification
is
ex-
p
ected.
E.6454
Consider
a
AB
C
D
E
F
GH
cub
e
of
edge
5
cm
inside
whic
h
the
AB
C
D
H
p
yramid
has
b
een
carv
ed.
1
a
Determine
the
measuremen
t
to
the
nearest
millime-
ter
of
segmen
t
[
AH
]
.
b
Determine
the
measuremen
t
to
the
nearest
millimeter
of
segmen
t
[
B
H
]
.
2
a
Giv
e
the
nature
and
dimensions
of
eac
h
of
its
faces.
b
Mak
e
a
pattern
of
the
p
yramid
AB
C
D
H
.
10.
Cones
:
volumes
E.9067
Prop
osition
:
A
cone
with
radius
r
and
heigh
t
h
has
a
v
olume
of
:
V
=
1
3
×
ı
×
r
2
×
h
Indication
:
w
e
will
use
:
ı
≈
3
;
1416
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7979
sacados/7979
A
B
C
D
E
F
G
H
I
J
K
Schéma
1
Schéma
2
Schéma
3
Schéma
4
chapExoCorrec/6454
sacados/6454
A
B
C
D
E
F
G
H
chapExoCorrec/9067
sacados/9067
r
Cône
hauteur
(
h
A
O
B
OA
cm
2
cm
A
O
S
S
3
cm
7
;
8
cm
3
cm
7
;
8
cm
S
A
H
A
H
C
C
A
S
O
I
Consider
a
cone
of
rev
olution
with
heigh
t
5
cm
and
base
radius
2
cm
.
P
oin
t
A
is
the
ap
ex
of
the
cone
and
O
is
the
cen
ter
of
its
base.
B
is
the
midp
oin
t
of
[
AO
]
.
Calculate
the
v
olume
of
the
cone
at
cm
3
.
Round
to
the
nearest
whole
n
um
b
er.
E.5457
A
marine
signal
buo
y
consists
of
t
w
o
cones
with
the
same
base,
as
sho
wn
in
the
figure
op-
p
osite.
The
follo
wing
dimensions
are
giv
en
:
S
O
=
0.8
m
;
S
O
=
0.5
m
;
O
A
=
0.25
m
Determine
the
total
v
olume
of
this
buo
y
rounded
to
the
nearest
dm
3
.
Note
:
w
e
will
use
:
ı
≈
3.1416
11.
Cones
:
pr
op
erties,
Pythagor
e
an
the
or
em
and
Thales
the
or
em
E.9108
Belo
w
is
a
cone
and
its
template
:
The
base
disc
has
a
radius
of
3
cm
and
an
y
segmen
t
of
the
lateral
surface
connecting
the
ap
ex
of
the
cone
to
a
p
oin
t
on
the
base
measures
7.8
cm
.
Determine
the
v
olume
of
this
cone
rounded
to
the
nearest
cubic
millimeter.
Hin
t
:
w
e
will
use
:
ı
≈
3.1416
E.9126
Consider
the
cone
with
base
C
,
cen-
ter
H
and
radius
[
H
A
]
,
and
ap
ex
S
suc
h
that
:
H
A
=
6
cm
;
H
S
=
15
cm
The
cone
is
in
tersected
b
y
a
plane
parallel
to
its
base
and
pasasn
t
through
the
p
oin
t
H
of
segmen
t
[
S
H
]
and
suc
h
that
:
S
H
=
5
cm
.
W
e
admit
that
:
the
section
is
a
circle
C
of
cen
ter
H
and
radius
[
H
A
]
where
the
p
oin
t
A
is
on
the
segmen
t
[
S
A
]
.
the
straigh
t
lines
(
A
H
)
and
(
AH
)
are
parallel.
Determine
the
measure
of
the
radius
of
the
circle
C
.
E.983
Consider
the
cone
sho
wn
opp
o-
site
with
v
ertex
S
and
whose
base
is
the
disk
of
radius
[
O
A
]
.
This
cone
has
heigh
t
S
O
=
8
cm
and
generatrix
S
A
=
10
cm
.
I
is
a
p
oin
t
on
segmen
t
[
S
O
]
suc
h
that
S
I
=
2
cm
.
Indication
:
w
e
will
use
:
ı
≈
3.1416
1
Sho
w
that
:
O
A
=
6
cm
.
2
Sho
w
that
the
exact
v
alue
of
the
v
olume
V
of
the
cone
is
equal
to
96
ı
cm
3
.
Giv
e
the
v
alue
rounded
to
the
nearest
mm
3
.
3
Determine,
to
the
nearest
degree,
the
measure
of
the
an-
gle
w
idehatAS
O
.
4
Cut
this
cone
with
a
plane
parallel
to
its
base
and
pass-
ing
through
p
oin
t
I
.
The
resulting
section
is
a
disk
with
cen
ter
I
,
a
reduction
of
the
base
disk.
a
Determine
the
ratio
k
of
this
reduction.
b
Let
V
‘
b
e
the
v
olume
of
the
cone
with
v
ertex
S
and
base
the
disk
with
cen
ter
I
.
Express
V
in
terms
of
V
,
then
giv
e
the
rounded
v
alue
of
V
to
the
nearest
mm
3
.
h
ttps:/
/c
hingmath.fr
A
O
B
OA
cm
2
cm
chapExoCorrec/5457
sacados/5457
A
O
S
S
chapExoCorrec/9108
sacados/9108
3
cm
7
;
8
cm
3
cm
7
;
8
cm
chapExoCorrec/9126
sacados/9126
S
A
H
A
H
C
C
chapExoCorrec/983
sacados/983
Brevet - 5,5 points
A
S
O
I
S
O
B
B
O
Niveau
maximum
de l’eau
O
A
H
A
H
C
C
O
O
A
A
S
g
r
C
1
C
2
¸
4
;
2
cm
7
cm
4
;
2
cm
7
cm
12.
Cones
:
enlar
gements
and
r
e
ductions
E.4204
In
practical
c
hemistry
w
ork,
studen
ts
use
con
tainers,
called
Erlenmey
er
flasks,
lik
e
the
one
sho
wn
b
elo
w
:
The
con
tainer
is
filled
with
w
ater
up
to
the
maxim
um
lev
el
indicated
on
the
diagram
b
y
an
arro
w.
Note
:
C
1
le
large
cone
with
v
ertex
S
et
base
the
disk
with
cen
ter
O
et
radius
O
B
;
C
2
the
small
cone
of
v
ertex
S
and
base
the
disk
of
cen
ter
O
et
of
radius
O
B
.
W
e
giv
e
:
S
O
=
12
cm
;
O
B
=
4
cm
Indication
:
w
e
will
use
:
ı
≈
3.142
1
The
v
olume
V
d
of
a
cone
of
rev
olution
of
radius
R
et
of
heigh
t
h
est
giv
en
b
y
the
form
ula
:
V
=
1
3
×
ı
×
R
2
×
h
Calculate
the
exact
v
alue
of
the
cone’s
v
olume
C
1
.
2
The
cone
C
2
is
a
reduction
of
the
cone
C
1
.
Giv
en
that
S
O
=
3
cm
.
a
What
is
the
co
efficien
t
of
this
reduction?
b
Pro
v
e
that
the
exact
v
alue
of
the
v
olume
of
the
cone
C
2
is
equal
to
ı
cm
3
.
3
a
Deduce
that
the
exact
v
alue
of
the
v
olume
of
w
ater
con
tained
in
the
con
tainer,
in
cm
3
,
is
63
ı
.
b
Giv
e
the
v
olume
of
w
ater,
rounded
to
the
nearest
cm
3
.
4
Is
this
v
olume
of
w
ater
greater
than
0.2
liters?
Explain
wh
y
.
E.9114
The
area
of
the
base
of
the
small
cone
of
rev
olution
is
5
mm
2
while
that
of
the
base
of
the
large
cone
is
80
mm
2
.
Calculate
the
area
enlargemen
t
co
effi-
cien
t.
Deduce
the
co
efficien
t
of
magnification
of
the
dimensions.
Kno
wing
that
the
v
olume
of
the
large
cone
is
384
mm
2
,
determine
the
v
olume
of
the
small
cone.
E.9121
The
area
of
the
small
base
is
27
m
2
.
The
area
of
the
large
base
108
m
2
.
The
heigh
t
of
the
large
cone
of
rev
olution
is
1
m
.
Calculate
the
heigh
t
of
the
small
cone
of
rev
olution.
13.
Cones
:
p
atterns
E.9107
Belo
w
is
the
pattern
of
a
cone
of
rev
olution
:
Express
the
measure
of
the
angle
¸
,
in
degrees,
in
terms
of
the
v
alues
of
r
and
h
.
E.11081
Belo
w
is
a
picture
of
a
cone
and
its
pat-
tern
:
The
base
disk
has
radius
3
cm
and
an
y
segmen
t
of
the
side
face,
connecting
the
ap
ex
of
the
cone
to
a
p
oin
t
on
the
base,
has
measure
7.8
cm
.
Hin
t
:
w
e’ll
use
ı
≈
3
;
1416
1
Determine
the
circumference
of
the
base
disk.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4204
sacados/4204
Pondichery Avril 2010
S
O
B
B
O
Niveau
maximum
de l’eau
chapExoCorrec/9114
sacados/9114
O
A
H
A
H
C
C
chapExoCorrec/9121
sacados/9121
O
O
A
A
S
chapExoCorrec/9107
sacados/9107
g
r
C
1
C
2
¸
sacados/11081
4
;
2
cm
7
cm
4
;
2
cm
7
cm
(
d
n
o
1
n
o
2
n
o
3
2
Deduce
the
angle
of
the
pattern
disk
section.
3
Determine
the
area
of
the
lateral
surface
of
this
cone.
14.
Complex
pr
oblems
and
tasks
E.7961
Heiata
and
Hiro
c
hose
as
their
w
edding
cak
e
a
pièce
mon
tée
consisting
of
3
cylindrical
cak
es
stac
k
ed
on
top
of
eac
h
other,
all
cen
tered
on
the
(
d
)
axis
as
sho
wn
b
elo
w
:
The
three
cylindrical
cak
es
are
the
same
heigh
t
10
cm
.
The
largest
cylindrical
cak
e,
the
n
o
1
,
has
radius
30
cm
.
The
radius
of
the
cak
e
n
o
2
is
equal
to
2
3
that
of
the
cak
e
n
o
1
.
The
radius
of
the
cak
e
n
o
3
is
equal
to
3
4
that
of
the
cak
e
n
o
2
.
1
Sho
w
that
the
radius
of
the
cak
e
n
o
2
is
20
cm
.
2
Calculate
the
radius
of
the
cak
e
n
o
3
.
3
Sho
w
that
the
exact
total
v
olume
of
the
moun
ted
piece
is
equal
to
15
250
ı
cm
3
.
Reminder
:
the
v
olume
V
of
a
cylinder
of
radius
R
and
heigh
t
h
is
giv
en
b
y
the
form
ula
V
=
ı
×
R
2
×
h
.
4
What
fraction
of
the
total
v
olume
is
the
v
olume
of
the
cak
e
n
o
2
?
Giv
e
the
result
as
an
irreducible
fraction.
E.6608
Belo
w
is
a
cylinder-shap
ed
c
ho
colate
b
o
x
:
Its
dimensions
are
:
a
heigh
t
of
8
cm
;
a
diameter
of
10
cm
.
T
o
close
this
b
o
x,
w
e
use
a
rubb
er
band
that
wraps
around
the
b
o
x
three
times.
On
eac
h
turn,
the
rubb
er
band
passes
through
the
cen
ters
of
the
t
w
o
disks.
Giv
e
the
length
of
the
rubb
er
band
when
it
is
placed
on
the
b
o
x
in
this
w
a
y
.
15.
Quotient
quantities
and
units
E.4207
A
t
the
construction
site
of
his
future
home,
Mr.
Dub
ois
encoun
ters
a
mason
who
seems
to
b
e
ha
ving
difficult
y
carrying
a
solid,
cylindrical
steel
ro
d.
This
ro
d
is
1.5
m
long
and
has
a
base
radius
of
4
cm
.
1
Calculate
the
v
olume
of
this
figure
rounded
to
the
nearest
cm
3
.
2
Steel
has
a
densit
y
of
7.85
g
=
cm
3
.
Calculate
the
mass
of
this
ro
d
rounded
to
k
g
.
Hin
t
:
w
e
will
use
:
ı
≈
3.142
E.9062
T
o
build
a
w
ell
in
his
garden,
Mr.
Martin
needs
to
buy
5
concrete
cylinders
lik
e
the
one
describ
ed
b
elo
w.
She
has
ro
om
in
her
trailer
for
5
cylinders,
but
she
can
only
transp
ort
a
maxim
um
of
500
k
g
.
Using
the
cylinder’s
c
haracteristics,
determine
the
minim
um
n
um
b
er
of
trips
M
me
Martin
needs
to
mak
e
to
transp
ort
his
5
cylinders
with
his
trailer.
Characteristics
of
a
cylinder
:
inner
diameter
:
90
cm
outer
diameter
:
101
cm
heigh
t
:
50
cm
densit
y
of
concrete
:
2
400
k
g
=
m
3
Determine
the
mass,
rounded
to
the
nearest
kilogram,
of
con-
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7961
sacados/7961
(
d
n
o
1
n
o
2
n
o
3
chapExoCorrec/6608
sacados/6608
chapExoCorrec/4207
sacados/4207
chapExoCorrec/9062
sacados/9062
Asie Juin 2019
1
;
70
m
4
;
40
m
7
;
5
cm
4
cm
12
cm
crete
required
for
the
design
of
this
w
ell.
Reminder
:
v
olume
of
the
cylinder
=
ı
×
radius
×
radius
×
heigh
t
Indication
:
w
e
will
use
:
ı
≈
3
;
14
E.6309
A
family
of
four
is
hesitating
b
et
w
een
t
w
o
p
o
ol
mo
dels.
They
are
gathering
information
to
help
them
mak
e
their
decision.
Information
1
:
The
t
w
o
p
o
ol
mo
dels
:
The
ˇroundı
p
o
ol
In
terior
heigh
t
:
1.20
m
T
op
view
:
a
circle
with
a
ra-
dius
of
1.70
m
The
ˇo
ctagonalı
p
o
ol
In
terior
heigh
t
:
1.20
m
View
from
ab
o
v
e
:
a
regular
p
olygon
with
an
outer
diame-
ter
of
4.40
m
.
Information
2
The
construction
of
an
ab
o
v
e-ground
p
o
ol
with
a
surface
area
of
less
than
10
m
2
do
es
not
require
an
y
administrativ
e
pro
ce-
dures.
Information
3
Recommended
minim
um
surface
area
p
er
swimmer
:
3.40
m
2
.
Information
4
Area
of
a
regular
o
ctagon
:
A
o
ctogone
=
2
2
×
R
2
where
R
is
the
radius
of
the
disk
outside
the
o
ctagon.
Information
5
Flo
w
rate
of
the
filling
tap
:
12
liters
of
w
ater
p
er
min
ute.
Indication
:
w
e
will
use
:
ı
≈
3.1416
1
Do
an
y
of
the
prop
osed
mo
dels
require
administrativ
e
pro
cedures?
2
The
four
mem
b
ers
of
the
family
w
an
t
to
swim
at
the
same
time.
Explain
wh
y
the
family
should
c
ho
ose
the
o
ctagonal
p
o
ol
in
this
case.
3
W
e
start
filling
this
o
ctagonal
p
o
ol
on
F
rida
y
at
14
h
00
and
lea
v
e
the
w
ater
running
o
v
ernigh
t
un
til
Saturda
y
morning
at
10
h
00
.
Will
the
p
o
ol
o
v
erflo
w?
16.
Unclassifie
d
exer
cises
E.4962
Belo
w
is
the
pattern
of
a
cone
of
rev
olu-
tion
:
Cut
out
the
pattern,
then
build
the
cone
of
rev
olution.
E.5691
In
this
exercise,
if
the
w
ork
is
not
completed,
lea
v
e
a
record
of
the
researc
h
an
yw
a
y
.
It
will
b
e
tak
en
in
to
accoun
t
in
the
assessmen
t.
A
m
uffin
pan
(for
b
aking)
consists
of
9
ca
vities.
All
of
these
ca
vities
are
iden-
tical.
Eac
h
ca
vit
y
is
shap
ed
lik
e
a
truncated
cone
(c
one
cut
by
a
plane
p
ar
al
lel
to
its
b
ase)
sho
wn
opp
osite.
The
dimensions
are
sho
wn
in
the
figure
opp
osite.
Reminders
:
V
olume
of
a
cone
with
base
radius
r
and
heigh
t
h
:
V
=
1
3
×
r
2
×
ı
×
h
1
‘
=
1
dm
3
Indications
:
w
e
will
use
:
ı
≈
3
;
1416
1
Sho
w
that
the
v
olume
of
a
ca
vit
y
is
appro
ximately
125
cm
3
.
2
Lea
has
prepared
1
liters
of
batter.
She
w
an
ts
to
fill
eac
h
ca
vit
y
of
the
mold
to
3
4
of
its
v
olume.
Do
es
she
ha
v
e
enough
batter
for
the
9
ca
vities
of
the
mold?
Justify
y
our
answ
er.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6309
sacados/6309
1
;
70
m
4
;
40
m
chapExoCorrec/4962
sacados/4962
chapExoCorrec/5691
sacados/5691
Brevet - Asie Juin 2013 Correction Catherine Royer
7
;
5
cm
4
cm
12
cm
P
etite
base
de
20
cm
de
diamètre
Grande
base
de
46
cm
de
diamètre
Hauteur
:
2
500
m
A
B
O
3
cm
A
O
S
A
O
C
K
E.7962
The
Rittershoffen
geothermal
p
o
w
er
plan
t
(Bas-R
hin)
w
as
in-
augurated
on
June
7
2016
.
A
w
ell
w
as
dug
to
capture
hot
w
ater
under
pressure,
at
a
depth
of
2
500
m
,
at
a
temp
erature
of
170
degrees
Celsius.
This
w
ell
is
shap
ed
lik
e
the
trun-
cated
cone
sho
wn
opp
osite.
The
prop
ortions
are
not
to
scale.
Note
:
the
v
olume
of
a
truncated
cone
is
calculatedâusing
the
follo
wing
form
ula
:
V
=
ı
3
×
h
×
R
2
+
R
×
r
+
r
2
where
h
is
the
heigh
t
of
the
truncated
cone,
R
is
the
ra-
dius
of
the
larger
base,
and
r
is
the
radius
of
the
smaller
base.
w
e
will
use
:
ı
≈
3.1416
1
Chec
k
that
the
v
olume
of
the
w
ell
is
appro
ximately
equal
to
225
m
3
.
2
The
earth
is
compacted
when
it
is
in
the
ground.
When
it
is
extracted,
it
is
no
longer
compacted
and
its
v
olume
increases
b
y
30
%
.
Calculate
the
final
v
olume
of
earth
to
b
e
stored
after
drilling
the
w
ell.
E.5676
The
figure
b
elo
w
represen
ts
a
solid
com-
p
osed
of
t
w
o
cones
of
rev
olution
sharing
the
same
base
disk,
whic
h
has
a
radius
of
measure
3
cm
.
The
distance
AB
measures
6
cm
.
Determine
the
v
olume
of
this
solid
rounded
to
the
nearest
cm
3
.
Hin
t
:
w
e’ll
use
:
ı
≈
3.1416
E.4961
Consider
the
cone
of
rev
olution
whose
base
radius
is
4
cm
and
the
hator
measures
4
cm
.
A
represen-
tation
of
this
solid
is
giv
en
b
elo
w
:
Determine
the
v
olume,
rounded
to
the
nearest
cubic
cen
time-
ter,
of
this
cone
of
rev
olution.
E.5440
Consider
an
hourglass
made
up
of
t
w
o
iden
tical
cones
of
the
same
v
ertex
C
and
whose
base
radius
is
AK
=
1.5
cm
.
T
o
protect
it,
it
is
enclosed
in
a
cylinder
of
heigh
t
6
cm
and
the
same
base
as
the
t
w
o
cones.
1
W
e
note
V
the
v
olume
of
the
cylinder
and
V
1
the
v
olume
of
the
hourglass.
All
v
olumes
will
b
e
expressed
in
cm
3
.
a
Sho
w
that
the
exact
v
alue
of
the
v
olume
V
of
the
cylin-
der
is
13.5
ı
.
b
Sho
w
that
the
exact
v
alue
of
V
1
is
4.5
ı
.
c
What
fraction
of
the
v
olume
of
the
cylinder
do
es
the
v
olume
of
the
hourglass
o
ccup
y?
(The
r
esult
wil
l
b
e
given
as
an
irr
e
ducible
fr
action)
.
Reminder
:
the
form
ula
for
the
v
olume
of
the
cone
is
:
aire
de
la
base
×
hauteur
3
2
W
e
put
6
cm
3
of
sand
in
to
the
hourglass.
Kno
wing
that
the
sand
will
flo
w
from
one
cone
to
another
with
a
flo
w
rate
of
80
cm
3
=
h
,
what
time
will
b
e
measured
b
y
this
hourglass?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/7962
sacados/7962
P
etite
base
de
20
cm
de
diamètre
Grande
base
de
46
cm
de
diamètre
Hauteur
:
2
500
m
chapExoCorrec/5676
sacados/5676
A
B
O
3
cm
chapExoCorrec/4961
sacados/4961
A
O
S
chapExoCorrec/5440
sacados/5440
A
O
C
K
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