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Let each edge of the square pyramid be = a units.
The diagonal on the square base = a√2 units
Half diagonal = a / √2 units

One lateral edge, half diagonal and the altitude (height) are in a right angle triangle.
     a² = (a/√2)² + (8√2)²
     a²/2 = 128
        a = 16 units.
Each of the four lateral surfaces are all equilateral triangles.
     So area of one face = 1/2 * √3/2 a * a = √3/4 * 256 = 110.85 square units.
 Total lateral surface area =  443.40  sq units

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