To determine the length of the diagonal of a cube with a side length of 3 cm, let's start from the basics and derive the formula for the space diagonal of a cube.
1. Understanding the Space Diagonal:
A cube has three dimensions equal, so if we have a side length \( a \), the space diagonal \( d \) derived through three-dimensional Pythagoras' theorem can be found.
2. Applying the Pythagorean Theorem in 3D:
Since the space diagonal spans from one vertex of the cube to the opposite vertex through the interior, its length can be described by:
[tex]\[
d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2}
\][/tex]
3. Simplifying the Formula:
Simplify the expression to:
[tex]\[
d = \sqrt{3} \cdot a
\][/tex]
4. Substituting the Side Length:
Now, we substitute the given side length of the cube, which is \( 3 \) cm:
[tex]\[
d = \sqrt{3} \cdot 3
\][/tex]
5. Calculating the Diagonal Length:
Simplify further to get:
[tex]\[
d = 3 \sqrt{3}
\][/tex]
6. Numeric Evaluation:
Compute the numeric value for \( 3 \sqrt{3} \). We can understand this in exact and decimal form. Since \( \sqrt{3} \approx 1.732 \):
[tex]\[
3 \sqrt{3} \approx 3 \times 1.732 = 5.196
\][/tex]
So, the side length calculation corresponds approximately to 5.196 cm.
7. Comparing with Given Choices:
To identify the exact matching from the given options, let's see their decimal equivalents:
- \( \sqrt{18} \approx 4.243 \)
- \( \sqrt{27} \approx 5.196 \)
- \( \sqrt{33} \approx 5.745 \)
- \( \sqrt{38} \approx 6.164 \)
From this, we see that the diagonal length of 5.196 cm precisely matches one of the given options.
8. Answer:
Hence, the length of the diagonal of a cube with a side length of 3 cm is:
[tex]\[
\boxed{\sqrt{27} \text{ cm}}
\][/tex]
