The diagonals of the three faces of a cuboid are x,y and z, respectively. What is the volume of the cuboid?
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Let a,b and c is the side of cuboid
root {a^2+b^2}=x
squaring both side
a^2+b^2=x^2.....................(1)
root {b^2+c^2}=y
squaring both side
b^2+c^2=y^2.....................(2)
root {a^2+c^2}=z
squaring both side
a^2+c^2=z^2.....................(3)
solve all equation we find
a^2=(x^2+z^2-y^2)/2
b^2=(x^2+y^2-z^2)/2
c^2=(y^2+z^2-x^2)/2
now
volume of cuboid=length * breath*height=abc=(a^2b^2c^2)^1/2
={(x^2+z^2-y^2)(x^2+y^2-z^2)(z^2+y^2-x^2)}^1/2/2root2
root {a^2+b^2}=x
squaring both side
a^2+b^2=x^2.....................(1)
root {b^2+c^2}=y
squaring both side
b^2+c^2=y^2.....................(2)
root {a^2+c^2}=z
squaring both side
a^2+c^2=z^2.....................(3)
solve all equation we find
a^2=(x^2+z^2-y^2)/2
b^2=(x^2+y^2-z^2)/2
c^2=(y^2+z^2-x^2)/2
now
volume of cuboid=length * breath*height=abc=(a^2b^2c^2)^1/2
={(x^2+z^2-y^2)(x^2+y^2-z^2)(z^2+y^2-x^2)}^1/2/2root2
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YES IT S CORRECT
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