Chapter 2.1, Problem 27PPSE

Solution

To Determine:

The relationship between the length of the edges and the area of each face.

The relationship between the area of one face and the surface area of the whole cube.

The surface area of the cube is $13.5{\text{cm}}^2$ .

The relationship between the length of the edges and the area of each face is

  $\text{Area}\text{of}\text{each}\text{face = }{\left(\text{length of the edges}\right)}^{\text{2}}$ .

The relationship between the area of one face and the surface area of the whole cube is

  $\text{Area}\text{of}\text{one}\text{face = }\frac{\text{ Surface area of the whole cube }}{\text{6}}$ .

Given:

Edges of cube length $=1.5 \text{cm}$

The surface area of the cube $=?$

Calculation:

Find the surface area:

A square's area, $a$ , is equal to the square of its edge length, $l$ . So, the face area is:

  $a=l^2$

Given that, $l=1.5\text{cm}$ , so

  $\begin{array}{l}a={\left(1.5\right)}^2\\ a=2.25\end{array}$

A cube has six square faces with equal edges, so its surface area, $s$ , is $6a$ . So, the Surface area is:

  $s=6a$

So, substitute $a=2.25$ in the equation $s=6a$ :

  $\begin{array}{l}s=6\times 2.25\\ s=13.5{\text{cm}}^2\end{array}$

Thus, the surface area of the cube is $13.5{\text{cm}}^2$ .

Find the relationship between the length of the edges and the area of each face:

In a cube, the area $F$ of each side with the length $a$ is as follows:

  $F=a^2\to \left(1\right)$

So, the relationship between the length of the edges and the area of each face of a cube will

be:

  $\text{Area}\text{of}\text{each}\text{face = }{\left(\text{length of the edges}\right)}^{\text{2}}$

Find the relationship between the area of one face and the surface area of the whole cube:

Consider

  $S=6a^2$

Substitute equation $\left(1\right)$

  $F=a^2$ in $S=6a^2$ :

  $\begin{array}{c}S=6F\\ F=\frac{S}{6}\end{array}$

Thus, the relationship between the area of one face and the surface area of the whole cube is:

  $\text{Area}\text{of}\text{one}\text{face = }\frac{\text{ Surface area of the whole cube }}{\text{6}}$