Question

Express the edge length of a cube as a function of the cube's diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length.

a. A function for the edge length of a cube is L(d) = _________________
b. A function for the surface area of a cube is Ald) = _____________
c. A function for the volume of a cube is V(d) = ________________

Answer 1

Explanation:

If L is the length of a side, the longest diagonal in the cube is

d = √(L² + L² + L²) = √(3L²) = L√3

meaning

a) L(d) = d/√3 = ⅓d√3

b) A(d) = 6(L(d))² = 6(⅓d√3)² = 6(⅟₉d²(3)) = 2d²

c) V(d) = (L(d))³ = (⅓d√3)³ = ⅟₂₇d³3√3 = ⅟₉d³√3