Right triangular prism

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

Correct answer:

S =  180 cm²

Step-by-step explanation:

$$\begin{gathered} a=12\text{ cm } \\ b=5\text{ cm } \\ c=4\text{ cm } \\ {u}_1=\sqrt{a^2+b^2}=\sqrt{12^2+5^2}=13\text{ cm } \\ {u}_2=\sqrt{a^2+c^2}=\sqrt{12^2+4^2}=4\sqrt{10}\text{ cm}\doteq 12.6491\text{ cm } \\ {u}_3=\sqrt{c^2+b^2}=\sqrt{4^2+5^2}=\sqrt{41}\text{ cm}\doteq 6.4031\text{ cm } \\ {S}_1=a\cdot b+c\cdot (a+b+u_1)=12\cdot 5+4\cdot (12+5+13)=180{\text{cm}}^2 \\ {S}_2=a\cdot c+b\cdot (a+c+u_2)=12\cdot 4+5\cdot (12+4+12.6491)\doteq 191.2456{\text{cm}}^2 \\ {S}_3=c\cdot b+a\cdot (c+b+u_3)=4\cdot 5+12\cdot (4+5+6.4031)\doteq 204.8375{\text{cm}}^2 \\ S=S_1=180{\text{cm}}^2 \end{gathered}$$

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