Total area

Calculate the total area (surface and bases) of a prism whose base is a rhombus which diagonals of 12cm and 18cm and prism height is 10 cm.

Correct result:

S =  648.6662 cm²

Solution:

$$\begin{gathered} d_1=12\text{ cm } \\ {d}_2=18\text{ cm } \\ h=10\text{ cm } \\ {S}_1={\displaystyle \frac{d_1\cdot d_2}{2}}={\displaystyle \frac{12\cdot 18}{2}}=108{\text{cm}}^2 \\ (d_1\mathrm{/}2{)}^2+(d_2\mathrm{/}2{)}^2=a^2 \\ a=\sqrt{(d_1\mathrm{/}2{)}^2+(d_2\mathrm{/}2{)}^2}=\sqrt{(12\mathrm{/}2{)}^2+(18\mathrm{/}2{)}^2}=3\sqrt{13}\text{ cm}\doteq 10.8167\text{ cm } \\ o=4\cdot a=4\cdot 10.8167=12\sqrt{13}\text{ cm}\doteq 43.2666\text{ cm } \\ {S}_2=o\cdot h=43.2666\cdot 10=120\sqrt{13}{\text{cm}}^2\doteq 432.6662{\text{cm}}^2 \\ S=2\cdot S_1+S_2=2\cdot 108+432.6662=648.6662{\text{cm}}^2 \end{gathered}$$

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