To solve the problem, we use key formulas related to rectangular prisms:

Step 1: Define Variables

Let (l), (w), (h) be the length, width, height of the prism.

Given:

• Total surface area (SA = 2(lw + lh + wh) = 80) cm² (assuming a typo, since 160 leads to an impossible negative value)
• Sum of dimensions: (l + w + h = 12) cm

Step 2: Calculate (lw + lh + wh)

From surface area:
(2(lw + lh + wh) = 80 \implies lw + lh + wh = 40)

Step 3: Find (l^2 + w^2 + h^2)

Using the identity:
((l + w + h)^2 = l^2 + w^2 + h^2 + 2(lw + lh + wh))

Substitute values:
(12^2 = l^2 + w^2 + h^2 + 2(40))
(144 = l^2 + w^2 + h^2 + 80)
(l^2 + w^2 + h^2 = 144 - 80 = 64)

Step 4: Compute Space Diagonal

Space diagonal (d = \sqrt{l^2 + w^2 + h^2} = \sqrt{64} = 8) cm

Answer: (\boxed{8}) cm. (Assuming the surface area was 80 cm² instead of 160 cm², as the original value is impossible.)

(\boxed{8})

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