One way is Cantor's diagonal argument. See link.

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Somehow I messed up andused 31 instead of 35

That is why you are receiving an improvement

Do a Google search for Diagonal of a cube

Find this website.

mathcentral.uregina.ca/QQ/database/QQ.09.04/brett1.HTML - Draw a cube

You will see how to find the diagonal of a cube.

You use Pythagorean Theorem

Draw a cube

Label each side as s

Draw the diagonal of the base of the cube

Diagonal of base = (s^2 + s^2 )^0.5

Let the diagonal of the base be the horizontal side of the right triangle whose hypotenuse is the diagonal of the cube.

The height of the cube is the vertical side of the right triangle whose hypotenuse is the diagonal of the cube.

Now determine the length of the diagonal of the cube.

(diagonal of base)^2 + (height of cube)^2 = (diagonal of cube )^2

Diagonal of base = (s^2 + s^2 )^0.5….Height of cube = s

Use Pythagorean Theorem

(diagonal of cube )^2 = (diagonal of base)^2 + (height of cube)^2

(diagonal of base)^2 = [(s^2 + s^2 )^0.5]^2 = s^2 + s^2

height of cube)^2 = s^2

(diagonal of cube)^2 = (s^2 + s^2 + s^2)

(diagonal of cube )^2 = (3 * s^2)

diagonal of cube = (3 * s^2)^0.5 = 35

(3 * s^2)^0.5 = 35

Square both sides

3s^2 = 31^2 = 1225

s^2 = 408.33

s = 40833^0.5

s = 20.2 cm