A hemispherical depression is cut-out from one face of a cubical wooden block such that the diameter l of the - Maths - Surface Areas and Volumes

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

Vimala · Accepted Answer

After cutting out a hemispherical depression, the view of the surface looks like this

The diameter of the hemisphere is $l$ units Thus, the side of the cube is also $l$ units The surface area of each of the other 5 faces of the cube is $l^{2} u n i t^{2}$ For the face from which a hemisphere has been cut, subtract the area of a circle of diameter $l$ units from $l^{2} u n i t^{2}$ and then add the area of a hemisphere of diameter $l$ units to it Area a circle of diameter $l$ units is $π \frac{l^{2}}{4} u n i t^{2}$ Surface Area of a hemisphere of diameter $l$ units is $2 π \frac{l^{2}}{4} u n i t^{2}$ That is the surface area of the face from which a hemisphere has been cut is: $l^{2} - \frac{π l^{2}}{4} + \frac{π l^{2}}{2} = l^{2} + \frac{π l^{2}}{4} u n i t^{2}$ Thus, the total surface area of the remaining solid is: $5 l^{2} + l^{2} + \frac{π l^{2}}{4} u n i t^{2} = 6 l^{2} + \frac{π l^{2}}{4} u n i t^{2}$ Therefore, the total surface area of the remaining solid is $6 l^{2} + \frac{π l^{2}}{4} u n i t^{2}$

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.