Missing Edge From Cube Surface Area

Idea

A cube has 6 congruent square faces. If the edge length is (a), then each face has area (a^2), so the total surface area is [ S = 6a^2. ] When the surface area is given and the edge is missing, solve this equation for (a).

Steps

1. Write the surface-area formula: [ S = 6a^2. ]
2. Substitute the given surface area for (S).
3. Isolate (a^2): divide both sides by 6.
4. Take the square root: [ a = \sqrt{\frac{S}{6}}. ] Use the positive root, since edge lengths are positive.
5. Simplify the radical or fraction if possible.

Check

To verify your answer, square the edge length and multiply by 6. The result should match the original surface area.

Example pattern

If (S = 96), then [ 6a^2 = 96 \Rightarrow a^2 = 16 \Rightarrow a = 4. ] Always make sure the final value has the correct units, such as cm or m.