Luas Limas Tegak dalam Kubus

19 Juni 10

Sebuah posting di yahoo answer menanyakan hal seperti ini:
6 buah limas tegak terbentuk di dalam kubus dengan rusuk 6 cm. Luas permukaan limas adalah..
a. (36sqrt{2}+36)cm^2
b. 72sqrt{2})cm^2
c. 36 cm^2
d. (36sqrt{2})cm^2

Hem mungkin yang menjadikan dia bingung adalah 6 limas masuk ke dalam sebuah kubus itu.. Ok illustrasinya adalah sebagi berikut:
limas-dalam-kubus
Dengan ilustrasi di atas maka akan tampak 6 buah limas yang kongruen masuk di dalam kubus, limas tersebut adalalah T.ABCD, T.BCGF, T.CGHD, T.DHEA, T.ABFE dan T.EFGH, semua limas tersebut memiliki tinggi TT’= setengah rusuk kubus = 3 cm. Nah sekarang kita mencari luas limasnya (salah satu saja)..
Perhatikan limas T.ABCD maka kita mencari dahulu tinggi segitiga ABT dengan phytagoras, yaitu:
t=sqrt{(TT prime)^2+(ST prime)^2}
t=sqrt{3^2+3^2}
t=sqrt{18}
t=3sqrt{2}
Sekarang kita hitung luas limas,
L limas = 4 x luas segitiga + luas persegi
L limas = 4 x L.ABT + L.ABCD
L limas = 4 x (setengah AB x ST) + L.ABCD
L limas = 4.({1/2}.6.3 sqrt{2}) + 6.6
L limas =36sqrt{2}+36

Jadi luas limas T.ABCD adalah (36sqrt{2}+36)cm^2

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