Question

The surface area of a cylinder which exactly fits in a cube of side $b$ is __________.

Answer 1

The total surface area of a cylinder is the sum of curved surface area and the area of two circular bases.

TSA $=$ Curved surface $+$ Area of Circular bases

TSA $=$$2\pi rh$ $+$ $2\pi r^2$

When the cylinder exactly fits in the cube of side “$b$”, the height equals to the edges of the cube and the radius equal to half the edges of a cube.

It means that,

$h=b$, and $r=\frac{b}{2}$

Then the TSA of a cylinder $=2\pi rh$ $+$ $2\pi r^2$

$=2\pi \left(\frac{b}{2}\right)b+2\mathrm{\pi }{\left(\frac{\mathrm{b}}{2}\right)}^2$

$=\pi b^2+\frac{\pi b^2}{2}$

$=\frac{3\pi b^2}{2}$.

Therefore, surface area of a cylinder which exactly fits in a cube of side $b$ is $\frac{3\pi b^2}{2}$.

Hence, the answer for the given blank is $\frac{3\pi b^2}{2}$.