How do the spacings of the threee planes `100`, `110`, and `111` of cube lattice vary?

Explanation

Applying the formula `d_(hkl) = (a)/(sqrt((h^(2) + k^(2) + l^(2)))`
`d_(100) = (a)/(sqrt((1^(2) + 0^(2) + 0^(2))))= a`
`d_(110) = (a)/(sqrt((1^(2) + 1^(2) + 0^(2))))= (a)/(sqrt2)`
`d_(111) = (a)/(sqrt(1^(2) + 1^(2) + 1^(2)) )= (a)/(sqrt3)`
Thus, `d_(100) : d_(110) : d_(111) : = 1 : (1)/(sqrt2) : (1)/(sqrt3) = 1 : 0.707 : 0.577`