SZABÁLYOS NÉGYOLDALÚ GÚLA – FELSZÍN, TÉRFOGAT, ÁTLÓS ÉS SÍKMETSZET KISZÁMÍTÁSA – SAMU MIHÁLY ÁLTALÁNOS ISKOLA PÉTERRÉVE
SZABÁLYOS NÉGYOLDALÚ GÚLA
1. A !"# #"#$%"& 1' (), *"+#"%"& 1 (). S-)/0+ 23 # !"# 4&"-/5%0. 2 At = a a = 12 cm,
a 2 2 2 2 10 = h a + 6
2
s = ha + 2
1
1
s = 10 cm,
At = 12
F = ? At = ?
At = 144 cm
2
F#
2
Pt = ? h a = ?
# '
------------
2
Pt = 2a ⋅ h a
F = At + Pt
Pt = 2 ⋅12 ⋅ 8
F = 144 + 192
Pt = 192 cm
F = 336 cm
2
2
100 = h a + 36 2
h a = 100 − 36 2
# '
h a = 64 2
OLDALA
ha =
64
h a = 8 cm
'. A !"# #"#$%"& 6 (), 0&0)### 7 (). S-)/0+ 23 # !"# 4&"-/5%0, /2)&0-&0%0 % 0%84*#00 a = 6 cm,
At = a
2
H = 4 cm,
At = 6
2
F = ? At = ?
2
F#
F#
At = 36 cm
Pt = ? h a = ?
a 2
Pt = 2 ⋅ a ⋅ h a
2
ha = H + 2
Pt = 2 ⋅ 6 ⋅ 5 Pt = 60 cm
h = 4 +3 2 a
2
H
2
2
h = 16 + 9 2 a
Ts = ? V = ?
h a = 25
# '
ha =
SÍKMETSZETE
2
Ts =
a ⋅H
V=
2 6⋅4
V=
2
Ts = 12cm
F = At + Pt
2
# '
---------------
Ts =
F = 96 cm
h a = 5 cm
3 36 ⋅ 4 3
V = 12 ⋅ 4
2
V = 48 cm
F = 36 + 60
25
At ⋅ H
3
2
9. S-)/0+ 23 #55#2 # !"5#2 # 0%84*#00, 0%84*#00, )&":5&2 0&0)### 9 (), 0"; )&0-&0%5&2 0&8<"&0& $&+3 '7 ()'. H = 30 cm, Td = 240 cm
Td =
2
d =a 2
At = a
2
16 = a 2
At = 8 2
1
1
V = ? At = ?
d ⋅H d ⋅ 30
240 =
2
H
d =?a =?
240 = d ⋅15
--------------+ '
d = 240 : 15
+ '
d = 16 cm
ÁTLÓS METSZETE
a=
16
a=
16 2
2
V=
(
⋅
2
)
2
V=
2
At = 64 ⋅ 2
2
At = 128 cm
2
At ⋅ H 3 128 ⋅ 30 3
V = 128 ⋅10 V = 1280 cm
3
2
a = 8 2 cm
7. A !"# *"+#"%"& 1' (), #"#$%"& = (). S-)/0+ 23 # !"# 0&0)##0 % *"+#")##0 *"+#")##0 s = 12 cm,
a s = h + 2 2 2 2 12 = h a + 8 2
a = 8 cm,
1
1
H = ? ha = ?
F#
---------# '
2
1
1
2
)
2
144 = H + 32 2
h = 144 − 64 2 a
#√' '
h = 80 2 a
OLDALA
(
12 = H + 4 2
H
2
2
2
2
144 = h a + 64 # '
a 2 s = H + 2
2
2 a
H = 144 − 32 2
#√' '
H = 112 2
ÁTLÓS METSZETE
h a = 16 ⋅ 5
H = 16 ⋅ 7
h a = 4 5 cm
H = 4 7 cm
>. A !"# #"#$%"& 1 (), *"+#")### $&+3 1 ()?8&" 5#:*@@ )350 0&0)###. 0&0)###. S-)/0+ 23 # !"# 4&"-/5%0 a = 10 cm, h a = H + 1 cm, F = ? At = ? Pt = ? h a = ? H = ?
At = a
2
At = 10
a 2
2
ha = H + 2
2
F#
F#
At = 100 cm
2
H
2
(H + 1) = H + 5 2
2
2
h a = H + 1 cm,
Pt = 2 ⋅ a ⋅ h a
h a = 12 + 1
Pt = 2 ⋅10 ⋅13
h a = 13 cm
Pt = 260 cm
H + 2H + 1 = H + 25 2
# '
2
F = At + Pt
2H = 25 - 1
---------------# '
F = 100 + 260
2H = 24
SÍKMETSZETE
F = 360 cm
H = 24 : 2 H = 12 cm
6. S-)/0+ 23 #- &:&5" %" !"# C#"#$%"*"+#"%" C#"#$%"*"+#"%" 4&"-/5%0, # P0 96 √9 ()'. a = s,
2
Pt = 36 3 cm , 2
# '
h = 2 a
a 2
2
a h = a − 2
2
2
# '
OLDALA
2 a
2
2
ha = a − 2
2
a
h =
4
4a − a 2
2 a
2
4
2
Pt = 2 ⋅ a ⋅ h a
2
a = ha +
F#
a = ? Pt = ? h a = ?
2
1
1
F = ? At = ?
----------------
a 2
s = ha +
2
3a
2
36 3 = 2 ⋅ a ⋅
4
ha =
3a
ha =
a 3
4
2
2
36 = a a=
2
a = 6 cm At = a At = 6
2
At = 36 cm
a 3
ha =
6 3
a 3 2
2 2
h a = 3 3 cm
36
2
ha =
F = At + Pt F = 36 + 36 3
(
)
F = 36 ⋅ 1 + 3 cm 2
2
2
SZABÁLYOS NÉGYOLDALÚ GÚLA – FELSZÍN, TÉRFOGAT, ÁTLÓS ÉS SÍKMETSZET KISZÁMÍTÁSA – SAMU MIHÁLY ÁLTALÁNOS ISKOLA PÉTERRÉVE
. S-)/0+ 23 # !"# 4&"-/5%0, # #"#$%"& 6 (), 0%84*#0# 7= () 9. a = 6 cm, V = 48 cm
3
F = ? At = ?
At = a
2
At = 6
2
V=
At = 36 cm
Pt = ? h a = ? H = ?
At ⋅ H
48 =
2
2
3
2
F#
F#
36 ⋅ H
h = 4 +3 2 a
H
Pt = 2 ⋅ a ⋅ h a
2
2
Pt = 2 ⋅ 6 ⋅ 5 Pt = 60 cm
2
F = At + Pt
h a = 25
F = 36 + 60
ha =
F = 96 cm
2
# '
H = 48 : 12
# '
SÍKMETSZETE
H = 4 cm
2
h a = 16 + 9 2
3
48 = 12 ⋅ H
----------------
a 2
ha = H +
25
2
h a = 5 cm
=. A !"# #"#$5#2 2&8<"&0&0 2&8<"&0&0 7= (), *"+#"%"& 1 (). S-)/0+ 23 # !"# 4&"-/5%0 K = 48 cm,
K = 4⋅a
s = 10cm,
48 = 4 ⋅ a
F = ? At = ?
a = 48 : 4
2
1
1 F#
a = 12 cm
Pt = ? h a = ? ------------
At = a
# '
2
At = 12
a 2 2 2 2 10 = h a + 6
2
Pt = 2 ⋅ a ⋅ h a
s = ha +
At = 144 cm
Pt = 2 ⋅12 ⋅ 8 Pt = 192 cm
h a = 100 − 36
2
2
F = At + Pt
h a = 64 2
# '
ha =
OLDALA
2
2
F = 144 + 192
64
F = 336 cm
h a = 8 cm
2
2
. S-)/0+ 23 # !"# 0%84*#00, 0%84*#00, # 4&"-/5& 96 () ', #- #"#$%"& $&+3 1 (). a = 10 cm,
At = a
2
F = 360 cm ,
At = 10
V = ? At = ?
At = 100 cm
2
2 2
Pt = ? h a = ?
2
F = At + Pt
Pt = 2 ⋅ a ⋅ h a
360 = 100 + Pt
260 = 2 ⋅10 ⋅ h a
Pt = 360 − 100
260 = 20 ⋅ h a
Pt = 260 cm
2
H =?
a 2 2 2 2 13 = H + 5
2
ha = H +
F#
F# H
2
V= V=
H = 169 − 25 2
h a = 260 : 20
H = 144
h a = 13 cm
------------
# '
3
V = 400 cm
H = 144
SÍKMETSZETE
3 100 ⋅12
V = 100 ⋅ 4
2
# '
At ⋅ H
3
H = 12 cm
1. A !"# P0 ' () ', 0&"& 4&"-/5& $&+3 196 ()'. S-)/0+ 23 # !"# #"#$%"%0. Pt = 72 cm ,
F = At + Pt
At = a
2
F = 360 cm ,
136 = At + 72
64 = a
2
a = ? At = ?
At = 136 − 72
a =
64
------------
At = 64 cm
a = 8 cm
2
2
2
2
11. A !"# P0>7 ()', #"#$5#2 2&8<"&0& $&+3 ' (). -)/0+ 23 # !"# 0%84*#00 Pt = 540 cm , 2
K = 72 cm, V = ? At = ? a = ? H = ? ha = ? ----------------
K = 4a
Pt = 2 ⋅ a ⋅ h a
72 = 4 ⋅ a
540 = 2 ⋅18 ⋅ h a
a = 72 : 4
540 = 36 ⋅ h a
a = 18 cm
2
H
h a = 540 : 36 # '
# '
At = 324 cm
V= V=
At ⋅ H 3 324 ⋅12 3
H = 225 − 81
V = 324 ⋅ 4
H = 144
V = 1296 cm
2
SÍKMETSZETE
2
2
2
2
2
At = 18
F#
F#
h a = 15 cm At = a
a 2 2 2 2 15 = H + 9 ha = H +
3
H = 144 H = 12 cm
2
1'. A !"# #"#$5#2 0";# ' √' (). H# #- *"+#"%" #- 0";J#" 7> °?* -&0 -8 @&, -)/0+ 23 # !"# 4&"-/5%0 % 0"; )&0-&0%5&2 0&8<"&0%0 d = 20 2 cm, α
= 45° cm,
d=a 2
d
20 2 = a 2
F = ? At = ?
H
a = ? Pt = ? h a = ? At = a
---------------d⋅H Td = 2 Td =
20 2 ⋅10 2 2
Td = 200 cm
2
2
At = 400 cm
2
+ '
ÁTLÓS METSZETE
2
(
h a = 10 2 2
=H
H = 10 2 cm + '
F#
F#
2
2
At = 20
2
2
=H
20 2
a = 20 cm
Td = ?H = ?
1
1
a 2
Pt = 2 ⋅ a ⋅ h a
2
ha = H +
H
) + 10 2
Pt = 2 ⋅ 20 ⋅10 3 2
Pt = 400 3 cm
h = 200 + 100 2 a
# '
# '
SÍKMETSZETE
h a = 300
F = At + Pt
h a = 100 ⋅ 3
F = 400 + 400 3
h a = 10 3 cm
F = 400 ⋅ 1 + 3 cm
2
(
)
2
SZABÁLYOS NÉGYOLDALÚ GÚLA – FELSZÍN, TÉRFOGAT, ÁTLÓS ÉS SÍKMETSZET KISZÁMÍTÁSA – SAMU MIHÁLY ÁLTALÁNOS ISKOLA PÉTERRÉVE
19. S-)/0+ 23 # !"# 4&"-/5%0, # #"#$%"& 7√' (), 0"; )&0-&0%5&2 )&0-&0%5&2 0&8<"&0& $&+3 9' () '. a = 4 2 cm,
d=a 2
Td = 32 cm , 2
Td =
d =4 2⋅ 2
F = ? At = ? Pt = ? h a = ? d =?H =?
d⋅H 2
32 =
d = 8 cm
32 = 4 ⋅ H
F#
F#
8⋅ H
d = 4⋅ 2
2
At = a
2
2
(
ha = 8 + 2 2 2
2
Pt = 2 ⋅ a ⋅ h a
2
(
At = 4 2
)
Pt = 2 ⋅ 4 2 ⋅ 6 2
2
Pt = 48 ⋅ 2
At = 16 ⋅ 2
)
2
At = 32 cm
h a = 64 + 8
H
2
Pt = 96 cm
2
h a = 72
# '
H = 8 cm
# '
ha =
SÍKMETSZETE
F = At + Pt
36 ⋅ 2
F = 32 + 96
h a = 6 2 cm
F = 128 cm
17. A !"# #"#$%"& 16 (), 0&0)### 9>?& #- *"+#")##5#2. S-)/0+ 23 # !"# 4&"-/5%0. a = 16 cm,
At = a
3
At = 16
H=
5
⋅ ha ,
2
F = ? At = ?
a h = H + 2 2 a
2
F#
F#
At = 256 cm
2
16h
3 2 h = ⋅ ha + 8 5
H
# '
9
h = 2 a
# '
SÍKMETSZETE
ha −
25 9
2
25
25
2 a
Pt = 2 ⋅ a ⋅ h a
2
25
2 a
Pt = ? h a = ? ----------------
25h a − 9h a
2
2
2
2
H=?
= 64
Pt = 2 ⋅16 ⋅10 Pt = 320 cm
= 64 / : 16
2
ha
⋅ h + 64 2 a
25
=4
F = 256 + 320
h a = 100 2
⋅ h 2a = 64
1>. S-)/0+ 23 # !"# /2)&0-&0%0, /2)&0-&0%0, # #"#$%"& 1 (), *"+#")### *"+#")### $&+3 19 () 2 a = 10 cm, a 2 2 ha = H + h a = 13 cm, 2 Ts = ? H = ?
F#
13 = H + 5 2
H
------------
2
Ts = Ts =
2
169 = H + 25 2
F = 576 cm
h a = 100
2
# '
# '
a ⋅H 2 10 ⋅12 2
Ts = 10 ⋅ 6
H = 169 − 25
Ts = 60 cm
2
H = 144 2
SÍKMETSZETE
H = 144 H = 12 cm
16. S-)/0+ 23 # !"# 4&"-/5%0 % 0%84*#00 0%84*#00 # #"#$%"& 6 (), /2)&0-&0& $&+3 1' ()'. a = 6 cm, Ts = 6 cm , 2
F = ?At = ? Pt = ? h a = ? V =?H =? ------------
Ts = 12 =
a ⋅H 2 6⋅H
a h = H + 2 2 a
F#
F#
2
2
h = 4 +3 2 a
H
2
2
2
At = a
F = At + Pt
2
At = 10
F = 100 + 60 2
F = 160 cm
At = 100 cm
12 = 3 ⋅ H
h a = 16 + 9
H = 12 : 3
h = 25
Pt = 2 ⋅ a ⋅ h a
ha =
25
Pt = 2 ⋅ 6 ⋅ 5
h a = 5 cm
Pt = 60 cm
H = 4 cm
2
# '
# '
SÍKMETSZETE
2 a
2
2
2
V= V= V=
2
F = At + Pt
/ ⋅ 25
h a = 10 cm
F#
2
2
H = 32 : 4
----------------
a h = H + 2 2 a
At ⋅ H 3 100 ⋅ 4 3 400 3
cm
3
2
2
