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V .Space oGe metry
: I T R A
P Polyhedra
s d r o
W Pronunciation Indonesian
e n i L w e k
S /skjuː laɪn / Garis Bers liangan n
o r d e h y l o
P /ˌpɒl ˈɪhɛdrən / Bidang Banyak a
r d e h y l o
P /ˌpɒl ˈɪhɛdrə/ Bidang Banyak j(amak) e
c a
F /feɪs / S isi e
d i
S / as ɪd/ S iis
e g d
E /ɛdʒ/ Rusuk
x e t r e
V /vɜ:teks/ Titiksudut s
e c i t r e
V /vɜ:tɪs :iz/ Titiksudut-titiksudut e
d i
S -edge / as ɪd ɛdʒ/ Rusuk tegak d
e s a
B -edge /beɪsd ɛdʒ/ Rusuk alas e
s a
B /beɪs/ Alas
l a r e t a
L Side /ˈlatərəl as ɪd/ Sis itegak m
s i r
P /ˈprɪz(ə)m/ Prisma m
s i r P t h g i
R / ar ɪtˈprɪz(ə)m/ Prisma tegak m
s i r P e u q il b
O /ə’bl:ik ˈprɪz(ə)m/ Prisma miring e
m u l o
V /ˈvɒ jluːm / I si a
e r A e c a f r u
S /ˈsəːfɪs ˈɛːrɪə/ Luas Permukaan e
c a f r u S d e v r u
C k/ əːv d ˈsəːfɪs/ Se ilmut ( tabung/kerucut) t
h g i e
H /haɪ /t Tinggi t
n a l
S Height / ls ɑːn ath ɪt / Panjang ruas garis Pelukis e
b u
C /kju:b/ Kubus d
i o b u
C /ˈkjuːbɔɪd / Balok e
c a
F -diagonal /feɪs dʌˈɪag(ə)n(ə l) / Diagona lbidang e
c a p
S -diagonal / esp ɪs dʌˈɪag(ə)n(əl) / Diagona lruang e
n a l P l a n o g a i
D /dʌˈɪag(ə)n(əl) epl ɪn / Bidang diagonal t
e
N /nɛt / Jaring-jaring d
i m a r y
P /ˈpɪrəmɪd / Limas d
il o S c i n o t a l
P /pləˈtɒnɪk ˈsɒlɪd / BangunpadatPlato n
o r d e h a r t e
T /ˌtɛtrəˈ hɛdrən / Tetrahedron n
o r d e h a x e
H /ˌhɛksəˈhɛdrən / Heksahedron n
o r d e h a c e d o
D /ˌdəʊdɛkəˈhɛdrən / Dodekahedron n
o r d e h a s o c
I /ˌʌɪkɒsəˈhɛdrən / Ikosahedron r
e d n il y
C /ˈsɪlɪndə/ Tabung e
n o
C /kəʊn / Kerucut e
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Example:
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1 SkewLines are nonintersecting ilnes thatare notpara lle .l
.
2 A polyhedron is the union o fpolygona lregions such that a ifnite .
r o i r e t n i e h t n i s e l o h y n a t u o h t i w d e s o l c n e s i e c a p s f o n o i g e r
.
3 Parts of a Polyhedra
.
a The polygona lshapes that e r a n o r d e h y l o p e h t m r o f
s t i d e ll a
c faces/sides. .
b The ilne segments where s t i e r a t e e m s e c a f o w t
s e g d
e .
.
c A point where three or a s i t e e m s e g d e e r o m
x e t r e v
e c a f x
e t r e v
e g d e
a s a H
e l o
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4 Types of a Polyhedra s
m s i r P . A
¬ A prism si a polyhedron that has identica lpolygona lfaces r
e h t o h c a e e t i s o p p
o .
¬ The segments that connect base side and top side is ca lled
e d i
s -edges .The others i s ca lled based-edges.
¬ The opposite ,identica lside o fa prism are ca lled its bases.
e h t d e ll a c e r a d n a s m a r g o l e ll a r a p e r a s e d i s r e h t o e h
T lateral
s e d i
s .
¬ I fthe latera lsides o fa prism are rectangles ,it is a right
. m s i r
p I fnot, i t i s ca lled an oblique prism .
¬ When naming a prism we use two main descriptors . First , e
w n e h t , e u q il b o r o t h g i r s i t i r e h t e h w y a s e
w say what type
l a r e t a l s t i f o r e b m u n e h t r o s e s a b s ’ m s i r p e h t m r o f n o g y l o p f o
. s e d i s
¬ Genera lly ,the f ormula f or volume o fprismi s t
h g i e h s t i × e s a b s t i f o a e r A = V
t n a l s e h t t o n t h g i e h l a e r e h t s i a l u m r o f s i h t n i t h g i e h e h T
. t h g i e h
m s i r p l a n o g a t n e p t h g i r
5 t h g i
r -sidedprism oobbililqquuee4re-csitdaendguplrairsmprism
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1 Cube
e b u C a f o s t r a P
6 lfat sides o f equa l se r a u q
s (ex :side ABCD )
12 edges o fequal l engths )B A e g d e : x e (
8 vertices ( ex :pointA)
12 f ace-diagonals ) F A e n il t n e m g e s : x e (
4 space-diagonals ) G A e n il t n e m g e s : x e (
6 diagona lplanes ) H G B A e d i s : x e (e b u C a f o t e
N Formula for Volume and o
a e r A e c a f r u
S f a Cube
s i e b u c f o e m u l o v r o f a l u m r o f e h T
) e g d e f o h t g n e l( =
V 3
s i a e r a e c a f r u s r o f d n A
) e g d e f o h t g n e l( × 6 =
A 2
)
2 Cuboid ( Rectangular Prism)
A rectangular prism has d il o s r a l u g n a t c e r 8e l g n
a s , 12 edges , equa l . s r u o f n i l e ll a r a p d n a
It is bounded by three t n e u r g n o c f o s r i a pn i g n i y l s e l g n a t c e r
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o a e r A e c a f r u S d n a e m u l o V r o f a l u m r o
F f a Cuboid
d i o b u c f o s e g d e t n e r e ff i d e e r h t e h t e m a n n a c e w t a h t e s o p p u S f o e m u l o v r o f a l u m r o f e h t , n e h T . t h g i e h d n a , h t d i w , h t g n e l s a s i d i o b u c t h g i e h × h t d i w × h t g n e l = V s i a e r a e c a f r u s r o f d n A ]) t h g i e h × h t d i w ( + ) t h g i e h × h t g n e l( + ) h t d i w × h t g n e l( [ × 2 = A .
B Pyramid
¬ A pyramid is a three-dimensiona lsoild with one polygona l e h t f o s e c i t r e v e h t g n i t c e n n o c s t n e m g e s e n il h t i w d n a e s a b . e s a b e h t e v o b a e r e h w e m o s t n i o p e l g n i s a o t e s a b
¬ The latera lsides o fa pyramid are triangles . I fthey are t s e l e c s o s
i riangles ,then i t i s a right pyramid ,otherwise i t i s n
a oblique pyramid.
¬ Genera lly ,the f ormula f orvolume o fpyramid i s
t h g i e h s t i × e s a b s t i f o a e r A × ) 3 / 1 ( = V e h t t o n t h g i e h l a e r e h t s n a e m o s l a a l u m r o f s i h t n i t h g i e h e h T . t h g i e h t n a l s d i m a r y p e r a u q s e u q il b
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5 The Platonic So ilds
r a l u g e r e l b i s s o p e v if y l n
O polyhedra exist .These ifve so ilds are S
D I L O S C I N O T A L P d e ll a c
r a l u g e R
n o r d e h y l o P
r e b m u N
s e d i S f o
h c a E
a s i e d i S
s n o g y l o P f o r e b m u N
x e t r e v a t a
n o r d e h a r t e
T 4 Triangle 3
n o r d e h a t c
O 8 Triangle 4
n o r d e h a s o c
I 2 0 Triangle 5
n o r d e h a x e
H 6 Square 3
n o r d e h a c e d o
D 1 2 Pentagon 3
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6 Cy ilnder
¬ A cy ilnder is a prismi nwhich the bases are circles orelilps.
¬ The volume o fa cyilnder is the area o fits base times its height
π = r2h
V
¬ The surface area o fa cyilnder i s π
+ π
=2 r2 2 rh
A
s u i d a r s ’ e s a b f o h t g n e l = r
r e d n il y c f o t h g i e h = h
¬ A cyilnder can be right or obilque ,and cyilnders are named in the s
d i m a r y p d n a s m s i r p s a y a w e m a s
. 7 Cone
¬ A cone is ilke a pyramid but with a circular base instead o fa .
e s a b l a n o g y l o p
¬ The volume o fa cone is one-third the area o fits base times its :
t h g i e h
π = 1 r2h
V 3
¬ The surface area o fa cone i s base surface area +curved surface :
a e r a
ro A =πr2 +πr r2 +h2 s u i d a r s ’ e s a b f o h t g n e l = r
e n o c f o h t g i e h = h
π + π
= r2 rs
A
r e d n il y C r a l u c r i C t h g i R
A AnObilqueElilptica lCyilnder
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e n o c f o t h g i e h t n a l s = s
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8 Sphere
¬ Sphere is the mathematica lword f or “bal.l” It i s the set o fal l r e t n e c e h t d e ll a c t n i o p n e v i g a m o r f e c n a t s i d d e x if a e c a p s n i s t n i o p
. e r e h p s e h t f o
¬ The volume o fa sphere i s :V = 4 rπ 3
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s e s i c r e x E s n o it s e u q e s e h t s r e w s n A . A .
1 Intersection o fthewalli n aclasscanbedescribedas_____________. .
2 Two planes i n aspace can ________________ or __________________ ,but cant’ . r e h t o h c a e _ _ _ _ _ _ _ _ _ _ _ _ .
3 What i sthenameo fobject thathas 6 vertices ,6 sides ,and 1 0 edges? .
4 I fa cuboid has edges whose lengths are 8 ,6 ,and 5 cm ,then its surface . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ s i a e r a .
5 A cyilndrica loi ltank can be if lled with 7,700 ilter gaso ilne .I fits base’s i l a i d a
r s70 cm ,then ifnd i ts height. .
6 Drawa neto fregularsquarepyramid and acone. .
7 Apyramid is i nscribed i n a cube .The top o fthe pyramid i s at the center o f s a h e b u c e h t f I . e b u c f o e d i s e s a b e h t s i e s a b s t i e li h w , e b u c f o e d i s p o t e h t . _ _ _ _ _ _ _ _ _ _ _ _ _ _ s i d i m a r y p e h t f o e m u l o v n e h t , m c 9 f o s e g d e .
8 A three dimensiona lobject that has 8 sides ,12 vertices ,and 1 8 _______ is . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ d e ll a c .
9 A cone is inscribed in a cyilnder .Their base side is equal ,whlie the top o f s a h r e d n il y c e h t f I . r e d n il y c f o p o t e h t f o r e t n e c e h t n i d e c a l p s i e n o c e h t e h t d n if , m c 4 2 f o t h g i e h d n a m c 7 f o i i d a
r curved surfaceareao fthecone.
. 0
1 What is the diameter o fthe sphere inscribed in a cube that is inscribed in ? 0 1 r e t e m a i d f o e r e h p s a . 1
1 The tota lsurface area o fa cube ,expressed in square centimeters ,is equa l i b u c n i d e s s e r p x e , e b u c e h t f o e m u l o v e h t o
t c centimeters .Compute the
a f o , s r e t e m i t n e c n i , h t g n e
l sideo fthesquare.
. 2
1 Two cy ilndrica lwater tank stand side by side .One has radius o f4 meters 3 f o s u i d a r a s a h r e h t o e h T . s r e t e m 5 . 2 1 f o h t p e d a o t r e t a w s n i a t n o c d n a u p s i r e t a W . y t p m e s i d n a s r e t e
m mped from the ifrst tank to the second
n u r p m u p e h t t s u m g n o l w o H . e t u n i m r e p s r e t e m c i b u c 0 1 f o e t a r a t a k n a t ? s k n a t h t o b n i e m a s e h t s i r e t a w e h t f o h t p e d e h t e r o f e b . 3
1 The pyramid ABCDE has a square base and al lfour triangular faces are e h T . l a r e t a li u q
