Question

Volume of cube with daigonal d

Answer 1

Answer:

Volume = $\frac{d^3}{3\sqrt{3} }$

Step-by-step explanation:

(Assume ABCD to be base square and EFGH corresponding upper square)

Let d = the length of diagonal |AG| and

let a = the length of each side of the cube.

Pythagoras theorem in triangle ACD:

$(AC)^2=(a)^2+(a)^2$

$AC = \sqrt{2}a$

Pythagoras theorem in triangle ACG:

$(AG)^2=(\sqrt{2}a)^2+(a)^2$      

$AG = \sqrt{3}a$

Now we can solve for a in terms of d:

$d=\sqrt{3} a$

$a=\frac{d}{\sqrt{3} }$

Therefore volume of the cube = $(\frac{d}{\sqrt{3} } )^3$ = $\frac{d^3}{3\sqrt{3} }$