Step 1: Understanding the Concept:
In any triangle, the smallest angle is always opposite the smallest side. First, we must evaluate or approximate the lengths of the given sides to identify the smallest one. Once identified, we apply the Law of Cosines to calculate the exact measure of the angle opposite to it.
Step 2: Key Formula or Approach:
1. Identify the shortest side, say $c$.
2. Law of Cosines: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$.
3. Standard trigonometric angle values to find angle $C$.
Step 3: Detailed Explanation:
Let's simplify and estimate the three sides to compare them:
Let side $a = 6 + \sqrt{12} = 6 + 2\sqrt{3}$. Since $\sqrt{3} \approx 1.732$, $a \approx 6 + 3.464 = 9.464$.
Let side $b = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \approx 4 \times 1.732 = 6.928$.
Let side $c = \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \approx 2 \times 2.45 = 4.9$.
Comparing the estimated values, $c$ is definitively the shortest side. Therefore, the smallest angle is angle $C$, which is opposite to side $c$.
Now we use the Cosine Rule to find $\cos C$:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
First, calculate the required square terms:
$a^2 = (6 + 2\sqrt{3})^2 = 36 + 24\sqrt{3} + 12 = 48 + 24\sqrt{3}$
$b^2 = (\sqrt{48})^2 = 48$
$c^2 = (\sqrt{24})^2 = 24$
Now calculate the numerator $a^2 + b^2 - c^2$:
\[ \text{Numerator} = (48 + 24\sqrt{3}) + 48 - 24 = 72 + 24\sqrt{3} \]
Factor out the common term 24:
\[ \text{Numerator} = 24(3 + \sqrt{3}) = 24\sqrt{3}(\sqrt{3} + 1) \]
Now calculate the denominator $2ab$:
\[ \text{Denominator} = 2(6 + 2\sqrt{3})(4\sqrt{3}) \]
\[ \text{Denominator} = 8\sqrt{3}(6 + 2\sqrt{3}) \]
\[ \text{Denominator} = 48\sqrt{3} + 16(3) = 48\sqrt{3} + 48 \]
Factor out the common term 48:
\[ \text{Denominator} = 48(\sqrt{3} + 1) \]
Substitute numerator and denominator back into the Cosine Rule equation:
\[ \cos C = \frac{24\sqrt{3}(\sqrt{3} + 1)}{48(\sqrt{3} + 1)} \]
The term $(\sqrt{3} + 1)$ cancels out nicely:
\[ \cos C = \frac{24\sqrt{3}}{48} = \frac{\sqrt{3}}{2} \]
We know from standard trigonometric tables that if $\cos C = \frac{\sqrt{3}}{2}$ in a triangle context ($0<C<180^\circ$), then $C = 30^\circ$, which in radians is $\frac{\pi}{6}$.
Step 4: Final Answer:
The smallest angle is $\frac{\pi}{6}$.