Question

7. A hemisphere is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the side of the cube. Determine the total surface area of the remaining solid?​

Answer 1

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Answer 2

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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

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Solution:

The figure below of the solid is created as per the given information along with the top view of the solid.

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter

From the figure, it’s clear that the surface area of the remaining solid includes TSA of the cube, CSA of the hemisphere, and excludes the base of the hemisphere.

Surface area of the remaining solid = TSA of the cubical part + CSA of the hemispherical part - Area of the base of the hemispherical part

The remaining area of the solid can be found by using the formulae;

TSA of the cube = 6 l2, where l is the length of the edge of the cube

CSA of the hemisphere = 2πr2

Area of the base of the hemisphere = πr2, where r is the radius of the hemisphere

Diameter of the hemisphere = Length of the edge of the cube = l

Radius of the hemisphere, r = l / 2

Surface area of the remaining solid = TSA of the cubical part + CSA of the hemispherical part - Area of the base of the hemispherical part

= 6 l2 + 2πr2 - πr2

= 6 l2 + πr2

= 6 l2 + π (l/2)2

= 6 l2 + πl2 / 4

= ¼ l2 (π + 24)

Thus, the surface area of the remaining solid is ¼ l2 (π + 24).