Certainly! Let's break down the problem step by step to determine the surface area of the cube.
1. Understanding the Given Information:
- The longest length between any two vertices of a cube is given as 3. This length is known as the space diagonal of the cube.
2. Relationship Between the Space Diagonal and the Side Length:
- For a cube, the space diagonal [tex]\( d \)[/tex] is related to the side length [tex]\( a \)[/tex] by the formula:
[tex]\[
d = a \sqrt{3}
\][/tex]
- Here, [tex]\( d = 3 \)[/tex]. We can use this relationship to solve for the side length [tex]\( a \)[/tex].
3. Solving for the Side Length [tex]\( a \)[/tex]:
- Rearrange the formula to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{d}{\sqrt{3}}
\][/tex]
- Substitute [tex]\( d = 3 \)[/tex] into the equation:
[tex]\[
a = \frac{3}{\sqrt{3}}
\][/tex]
- Simplifying [tex]\(\frac{3}{\sqrt{3}}\)[/tex]:
[tex]\[
a = \sqrt{3}
\][/tex]
4. Calculating the Surface Area of the Cube:
- The surface area [tex]\( SA \)[/tex] of a cube is calculated using the formula:
[tex]\[
SA = 6a^2
\][/tex]
- We have already determined [tex]\( a = \sqrt{3} \)[/tex]. Substitute this value into the surface area formula:
[tex]\[
SA = 6(\sqrt{3})^2
\][/tex]
- Simplifying [tex]\( (\sqrt{3})^2 \)[/tex]:
[tex]\[
SA = 6 \cdot 3
\][/tex]
- Thus, the surface area is:
[tex]\[
SA = 18
\][/tex]
Therefore, the surface area of the cube is [tex]\( 18 \)[/tex] square units.
