Cube triangle volume

In the cube ABCDEFGH, the area of triangle ABK is √20 cm². How much cm² is the volume of ABGH in a cube if you know that K is the midpoint of edge CG?

Final Answer:

S =  11.3137 cm²

Step-by-step explanation:

$$\begin{gathered} S_1=\sqrt{20}=2\sqrt{5}{\text{cm}}^2\doteq 4.4721{\text{cm}}^2 \\ b=\sqrt{a^2+(a/2{)}^2} \\ b=\sqrt{5}/2\cdot a \\ {S}_1=\frac{a\cdot b}{2} \\ {S}_1=\frac{a\cdot \sqrt{5}/2\cdot a}{2} \\ 4S_1=\sqrt{5}{a}^2 \\ a=\sqrt{4\cdot S_1/\sqrt{5}}=\sqrt{4\cdot 4.4721/\sqrt{5}}=2\sqrt{2}\text{cm}\doteq 2.8284\text{cm} \\ d=\sqrt{2}\cdot a=\sqrt{2}\cdot 2.8284=4\text{cm} \\ S=a\cdot d=2.8284\cdot 4=8\sqrt{2}\doteq 11.3137{\text{cm}}^2 \\ \text{ Verifying Solution: } \\ b=\sqrt{a^2+(a/2{)}^2}=\sqrt{2.8284^2+(2.8284/2{)}^2}=\sqrt{10}\text{cm}\doteq 3.1623\text{cm} \\ {S}_1=\frac{a\cdot b}{2}=\frac{2.8284\cdot 3.1623}{2}=2\sqrt{5}{\text{cm}}^2\doteq 4.4721{\text{cm}}^2 \end{gathered}$$

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