Question

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

Answer 1

Let the edge of the cube = a.

The diameter of the hemispherical depression is equal to the edge of the cube.

= r = `a/2`

1) Total surface area of the cube

`TSA _"cube" ​= 6a^2`

2) Area removed from one face of the cube

A circular portion is cut out.

Area of circle = `πr^2`

= `π(a/2)^2`

= `(πa^2)/4`

3) Curved surface area of the hemispherical depression (added)

CSA of hemisphere = 2πr²

= `2π(a/2)^2`

= `(πa^2)/2`

Surface area of the remaining solid

`6a^2 − (πa^2)/4 + (πa^2)/2`

= `6a^2 + (πa^2)/4`

Surface Area = `(a^2(24 + π))/4` square units`