Cube

The volume of a cube is given by $l^3$ , where $l$ denotes its length.Since we are given that the voume is $512\;\text{cm}^3$ ,hence $l^3=512\;\text{cm}^3\implies l=\boxed{8\;\text{cm}}$ .

The surface area of a cube is given by $6l^2$ .We want to find half of that,so we have to find $3l^2$ .Since we know that $l=8\;\text{cm}$ ,we can simply plug that in to get $3l^2=3(8\;\text{cm})^2=\boxed{192\;\text{cm}^2}$

Relevant wiki: Surface Area - Problem Solving

The volume of a cube is the cube of its edge. Let $a$ be the edge length, then

$512=a^3$ $\color{#D61F06}\implies$ $a=8$

Half of its surface area is

$s=\dfrac{1}{2}(6)(8^2) = \color{#69047E}\boxed{192~cm^2}$

Area of cube is given by $a^3$ .
$a^3 = 512\\ a = 8$
Half of its surface area = $3a^2 = 3×8^2 = 3×64 = \color{#69047E}{\boxed{192}}$ .