Question

the square on the diagonal of a cube has an area of 192 cm square calculate the side of the cube and the total surface area of the cube

Answer 1

Side of the cube = S
Length of diagonal on one face of cube = L
Length of diagonal of the cube = Z

We know that,
$l = s \sqrt{2}$
$z = s \sqrt{3}$
(See derivation of these formulae in attached photo)

Given Z²=192 cm²
Squaring the equation of Z on both sides we get,
${z}^{2} = {(s \sqrt{3} )}^{2}$
192=S²×3
S²=192/3 = 64
S=4 cm
Total surface area of cube=6S²
TSA=6×4²=6×64= 384 cm²

Hence side of given cube = S = 4cm
and Total surface area of cube =TSA=384 cm²

Answer 2

Answer:

side of cube = 8 cm

total surface area = 384 cm²

Step-by-step explanation:

diagonal of cube = D

side of cube = S

D = S√3 (equation for diagonal of cube)

D² = 192cm²

∴192 = (S√3)²

192 = S² × 3

S² = 64

S = √64 = 8

total surface area = 6S²

total surface area = 6 × 8² = 6 × 64

total surface area = 384cm²