Rhombus base

Calculate the volume and surface area of prisms whose base is a rhombus with diagonals u₁ = 12 cm and u₂ = 15 cm. Prism height is twice base edge length.

Correct result:

V =  1728.84 cm³
S =  918 cm²

Solution:

$$\begin{gathered} u_1=12\text{ cm } \\ {u}_2=15\text{ cm } \\ (u_1\mathrm{/}2{)}^2+(u_2\mathrm{/}2{)}^2=a^2 \\ a=\sqrt{(u_1\mathrm{/}2{)}^2+(u_2\mathrm{/}2{)}^2}=\sqrt{(12\mathrm{/}2{)}^2+(15\mathrm{/}2{)}^2}\doteq 9.6047\text{ cm } \\ {S}_1={\displaystyle \frac{u_1\cdot u_2}{2}}={\displaystyle \frac{12\cdot 15}{2}}=90{\text{cm}}^2 \\ h=2\cdot a=2\cdot 9.6047=3\sqrt{41}\text{ cm}\doteq 19.2094\text{ cm } \\ V=S_1\cdot h=90\cdot 19.2094=270\sqrt{41}=1728.84{\text{cm}}^3=1.7\cdot 10^3{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} o=4\cdot a=4\cdot 9.6047=6\sqrt{41}\text{ cm}\doteq 38.4187\text{ cm } \\ {S}_2=o\cdot h=38.4187\cdot 19.2094=738{\text{cm}}^2 \\ S=2\cdot S_1+S_2=2\cdot 90+738=918{\text{cm}}^2 \end{gathered}$$

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