Question

diagonal of three faces of cuboid is equal to x,y,z . find the volume of cuboid?

Answer 1

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 sides
b^2+c^2=y^2 ---------------(2)

root {a^2+c^2}=z
squaring both sides
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 x breadth x height =abc =(a^2b^2c^2)^1/2

={(x^2+z^2-y^2)(x^2+y^2-z^2)(z^2+y2-x^2)}^1/2/2root2