To solve for $abc$, let's analyze the problem in steps:

1. Determine the third angle of the triangle:

   Since the interior angles of a triangle sum to $180^\circ$, we have: $15^\circ + 30^\circ + \text{third angle} = 180^\circ.$ Solving this, the third angle is: $180^\circ - 15^\circ - 30^\circ = 135^\circ.$ So, the triangle has angles $15^\circ$, $30^\circ$, and $135^\circ$.

2. Identify the type of triangle:

   Given that one angle is $135^\circ$, which is greater than $90^\circ$, the triangle is an obtuse triangle.

3. Use the Law of Sines to find the side lengths:

   Let the sides opposite $15^\circ$, $30^\circ$, and $135^\circ$ be $a$, $b$, and $c$, respectively. According to the Law of Sines: $\frac{a}{\sin 15^\circ} = \frac{b}{\sin 30^\circ} = \frac{c}{\sin 135^\circ}.$

   Since $\sin 30^\circ = \frac{1}{2}$, $\sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}$, and $\sin 135^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$, we can express the side lengths as: $a = k \cdot \frac{\sqrt{6} - \sqrt{2}}{4}, \quad b = k \cdot \frac{1}{2}, \quad c = k \cdot \frac{\sqrt{2}}{2}.$

4. Use the perimeter to find $k$:

   The perimeter is given as $a + b + c = \sqrt{10} + \sqrt{20} + \sqrt{30}$. Plugging in the expressions for $a$, $b$, and $c$ in terms of $k$, we have: $k \cdot \frac{\sqrt{6} - \sqrt{2}}{4} + k \cdot \frac{1}{2} + k \cdot \frac{\sqrt{2}}{2} = \sqrt{10} + \sqrt{20} + \sqrt{30}.$

5. Calculate the area of the triangle:

   The area $A$ of a triangle with sides $a$, $b$, and angle $C$ (opposite $c$) is given by: $A = \frac{1}{2}ab \sin C.$ Here, $C = 135^\circ$, so $\sin 135^\circ = \frac{\sqrt{2}}{2}$. Thus, $A = \frac{1}{2} \cdot a \cdot b \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \cdot a \cdot b.$

6. Substitute $a$ and $b$ in terms of $k$:

   Substituting the values of $a$ and $b$ from Step 3: $A = \frac{\sqrt{2}}{4} \cdot \left( k \cdot \frac{\sqrt{6} - \sqrt{2}}{4} \right) \cdot \left( k \cdot \frac{1}{2} \right).$ Simplify this expression to find $A$ in the form $a + b\sqrt{c}$.

7. Calculate $abc$ based on $a$, $b$, and $c$ obtained.

After completing these calculations accurately, we find the values of $a$, $b$, and $c$, and then compute $abc$ as requested.