Diagonal only

Geometry Level 3

What is the surface area of cube that has a space diagonal of 100 100 ?


The answer is 20000.

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1 solution

By the pythagorean theorem , we have

y = a 2 + a 2 = 2 a 2 = a 2 y=\sqrt{a^2+a^2}=\sqrt{2a^2}=a\sqrt{2}

By the pythagorean theorem again, we have

10 0 2 = a 2 + ( a 2 ) 2 100^2=a^2+(a\sqrt{2})^2 \color{#D61F06}\large \implies 10000 = a 2 + 2 a 2 = 3 a 2 10000=a^2+2a^2=3a^2 \color{#D61F06}\large \implies a 2 = 10000 3 a^2=\dfrac{10000}{3}

Finally, the surface area is

A = 6 a 2 = 6 ( 10000 3 ) = A=6a^2=6\left(\dfrac{10000}{3}\right)= 20000 \color{#3D99F6}\boxed{\large 20000}

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