Calculate Face Diagonals: Rectangular Prism with Dimensions 4×5×7

We will use the Pythagorean theorem to find diagonal AD1:

$AA_1+A_1D_1=AD_1$

Let's input the known data:

$5^2+7^2=D_1A^2$

$D_1A^2=25+49=74$

Let's find the square root:

$AD_1=\sqrt{74}$

From the data we can see that:

$AA_1=DD_1=5$

Now let's look at triangle DD1C1 and calculate DC1 using the Pythagorean theorem:

$D_1D^2+D_1C_1^2=C_1D^2$

Let's input the existing data:

$5^2+4^2=C_1D^2$

$C_1D^2=25+16=41$

Let's find the square root:

$DC_1=\sqrt{41}$

Now let's focus on triangle A1D1C1 and find diagonal A1C1:

$A_1D_1^2+D_1C_1^2=A_1C_1^2$

Let's input the known data:

$7^2+4^2=A_1C_1^2$

$A_1C_1^2=49+16=65$

Let's find the square root:

$A_1C_1=\sqrt{65}$

Now we have all 3 lengths of all possible diagonal corners in the box:

$\sqrt{74},\sqrt{41},\sqrt{65}$