Cuboid face diagonals

The lengths of the cuboid edges are in the ratio 1: 2: 3. Will the lengths of its diagonals be the same ratio?

The cuboid has dimensions of 5 cm, 10 cm, and 15 cm. Calculate the size of the wall diagonals of this cuboid.

Result

y =  0

Solution:

a=5 b=10 c=15  a:b:c=1:2:3  u1=a2+b2=52+102≐5 5≐11.1803 u2=a2+c2=52+152≐5 10≐15.8114 u3=b2+c2=102+152≐5 13≐18.0278  k1=u1:u2=11.1803:15.8114≐0.7071 k2=u3:u2=18.0278:15.8114≐1.1402 k3=u3:u1=18.0278:11.1803≐1.6125  k1,k2,k3≠1,2,3  y=0a=5 \ \\ b=10 \ \\ c=15 \ \\ \ \\ a:b:c=1:2:3 \ \\ \ \\ u_{1}=\sqrt{ a^2+b^2 }=\sqrt{ 5^2+10^2 } \doteq 5 \ \sqrt{ 5 } \doteq 11.1803 \ \\ u_{2}=\sqrt{ a^2+c^2 }=\sqrt{ 5^2+15^2 } \doteq 5 \ \sqrt{ 10 } \doteq 15.8114 \ \\ u_{3}=\sqrt{ b^2+c^2 }=\sqrt{ 10^2+15^2 } \doteq 5 \ \sqrt{ 13 } \doteq 18.0278 \ \\ \ \\ k_{1}=u_{1}:u_{2}=11.1803:15.8114 \doteq 0.7071 \ \\ k_{2}=u_{3}:u_{2}=18.0278:15.8114 \doteq 1.1402 \ \\ k_{3}=u_{3}:u_{1}=18.0278:11.1803 \doteq 1.6125 \ \\ \ \\ k_{1},k_{2},k_{3} \ne 1,2,3 \ \\ \ \\ y=0

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