Weird solid

Note that we can separate the solid into the following solids, each with breadth $10$ :

• A rectangular prism with length $12$ and width $11$

• Two rectangular prisms with length and width $3$

• A triangular prism with an equilateral triangle base, with side length $4$ .

Hence, the volume is:

$\begin{aligned} V &= 10\left(12 \times 11 + 3^2 + \dfrac{4^2\sqrt{3}}{4}\right)\\ V &= 10(150 + 4\sqrt{3})\\ V &= 1500 + 40\sqrt{3}\\ V &\approx \boxed{1569.282 \text{ cubic units.}} \end{aligned}$

$volume=[(0.5 \times 4^2 \times \sin 60) + (12 \times 11) + (2 \times 3^2)] \times 10 \approx \boxed{1569.282}$