Right triangular prism

We have cuboid with a base and dimensions of 12 cm and 5 cm and height of 4 cm. The tablecloth cut it into two identical triangular prisms with right triangular bases. The surface of the created prisms was painted with color. Calculate the surface area of one of these two triangular prisms.

Correct result:

S =  180 cm²

Solution:

$a=12 \ \text{cm} \ \\ b=5 \ \text{cm} \ \\ c=4 \ \text{cm} \ \\ \ \\ u_{1}=\sqrt{ a^2+b^2 }=\sqrt{ 12^2+5^2 }=13 \ \text{cm} \ \\ u_{2}=\sqrt{ a^2+c^2 }=\sqrt{ 12^2+4^2 } \doteq 4 \ \sqrt{ 10 } \ \text{cm} \doteq 12.6491 \ \text{cm} \ \\ u_{3}=\sqrt{ c^2+b^2 }=\sqrt{ 4^2+5^2 } \doteq \sqrt{ 41 } \ \text{cm} \doteq 6.4031 \ \text{cm} \ \\ \ \\ S_{1}=a \cdot \ b + c \cdot \ (a+b+u_{1})=12 \cdot \ 5 + 4 \cdot \ (12+5+13)=180 \ \text{cm}^2 \ \\ S_{2}=a \cdot \ c + b \cdot \ (a+c+u_{2})=12 \cdot \ 4 + 5 \cdot \ (12+4+12.6491) \doteq 191.2456 \ \text{cm}^2 \ \\ S_{3}=c \cdot \ b + a \cdot \ (c+b+u_{3})=4 \cdot \ 5 + 12 \cdot \ (4+5+6.4031) \doteq 204.8375 \ \text{cm}^2 \ \\ \ \\ S=S_{1}=180 \ \text{cm}^2$

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