A sphere and a cube

$\quad \quad \\ \quad Given\quad the\quad the\quad surface\quad areas\quad are\quad equal\\ \quad \quad 4\pi { r }^{ 2 }\quad =\quad 6{ a }^{ 2 }\\ \Rightarrow \quad \sqrt { \frac { 2\pi }{ 3 } } r\quad =\quad a\\ Ratio\quad of\quad volumes\quad is:\\ \frac { 4\pi { r }^{ 3 } }{ 3 } :\quad { a }^{ 3 }\\ \Rightarrow \frac { 4\pi { r }^{ 3 } }{ 3 } :\frac { 2\pi }{ 3 } \sqrt { \frac { 2\pi }{ 3 } } { r }^{ 3 }\\ \Rightarrow \quad \quad 2\quad :\sqrt { \frac { 2\pi }{ 3 } } \\ \Rightarrow \sqrt { 6 } :\quad \sqrt { \pi }$

$\color{#3D99F6}{\text{Given that Surface Area of Sphere = Surface Area of Cube}}$

$4 \pi r^2 = 6 a^2$

$\frac{r}{a}=$ $\sqrt {\frac{3}{2 \pi}}$

$\Rightarrow r = a\sqrt {\frac{3}{2 \pi}}$

$\text{Volume of Sphere= }$ $\frac{4}{3} \pi r^3 = \frac{4}{3} \pi {a\sqrt {\frac{3}{2 \pi}}}^3 = 2a^3 \sqrt{\frac{3}{2 \pi}}$

$\text{Volume of Cube}= a^3$

$\text{Required Ratio}=$ $\large \frac{2a^3 \sqrt{\frac{3}{2 \pi}}}{a^3} = 2 \sqrt{\frac{3}{2 \pi}} = \sqrt{\frac{12}{2\pi}} = \boxed{\color{#20A900}{\sqrt 6 : \sqrt \pi}}$