Question

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Answer 1

It is given to us that diameter of Hemisphere is equal to edge of cube = l

$\Rightarrow$ Radius of Hemisphere $=r=\frac{l}{2}$

Surface Area of solid = Surface Area of 5 faces of cube + Inner Surface Area of Hemisphere + Blue Area on the Top Face ABCD

$=5.l^2+2.\pi r^2$ + ( Area of Square ABCD - Area of Inscribed Circle in square ABCD)

$=5l^2+2.\pi .{\left(\frac{l}{2}\right)}^2+(l^2-\pi .(\frac{l}{2}{)}^2)$

$=5l^2+\pi .\frac{l^2}{2}+l^2-\pi .\frac{l^2}{4}$

$=6l^2+\pi .\frac{l^2}{4}$

$=\frac{(24.l^2+\pi .l^2)}{4}$

$=\frac{l^2(\pi +24)}{4}$