Math Problem Solver

The surface area of a cube is $384\, \text{cm}^2$.

Note: Approximate solutions are rounded to the nearest hundredths place.

Given: $$\begin{aligned}V&=512\end{aligned}$$

Surface area

The surface area of a cube is given by the equation: $$\begin{aligned}S&=6 s^{2}\end{aligned}$$ where $S$ is surface area and $s$ is side.

Need:$$\begin{aligned}V&=s^{3}\end{aligned} \quad \Rightarrow \quad \begin{aligned}512&=s^{3}\end{aligned} \quad \Rightarrow \quad \begin{aligned}s&=8\end{aligned}$$

Solution:$$\begin{aligned}S&=6 s^{2}\\S&=6 \left(8\right)^{2}\\S&=384\end{aligned}$$

A cube is a three dimensional solid with six congruent square faces, eight vertices, and twelve edges. All angles are right angles. Any square face can be the base, and the parallel face is the top. Each side has a length of s. Less commonly, one pair of parallel sides of the base is the length l, the other pair is the width w, and the sides connecting the base to the top are the height h.

The diagonal d of a cube is the distance between opposite vertices on different faces. $$\begin{aligned} d &= s \sqrt{3} = \sqrt{s^{2} + s^{2} + s^{2}} \\ &= \sqrt{l^{2} + w^{2} + h^{2}} \end{aligned}$$

The base perimeter P_B of a cube is the distance around the base. $$\begin{aligned} P_{B} &= 4 s \\ &= 2 l + 2 w \end{aligned}$$

The base area A_B of a cube is the region enclosed inside the base. $$\begin{aligned} A_{B} &= s^2 \\ &= l w \end{aligned}$$

The lateral surface area A_L of a cube is the sum of areas of its square faces, excluding the areas of the top and the base faces. $$\begin{aligned} A_{L} &= 4 s^{2} \\ &= 2 (l + w) h \end{aligned}$$

The perimeter P of a cube is the sum of all the sides.$$\begin{aligned} P &= 12 s \\ &= 4 (l + w + h) \\ &= 4 \sqrt{3} d\end{aligned}$$

The surface area S of a cube is the region occupied by all its faces. $$\begin{aligned} S &= 2 A_{B} + A_{L} \\ &= 6 s^{2} \\ &= 2 (l w + l h + w h) \\ &= 2 d^{2} \end{aligned}$$

The volume V of a cube is the amount of space it occupies. $$\begin{aligned} V &= A_{B} h \\ &= s^{3} \\ &= l w h \\ &= \frac{d^{3} \sqrt{3}}{9} \end{aligned}$$

Classification: Polyhedron, Prism, Platonic Solid