# 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

**Solution:**

The figure below of the solid can be created as per the given information with the top view of the solid.

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 formulae;

TSA of the cube = 6 l^{2}, where l is the length of the edge of the cube

CSA of the hemisphere = 2πr^{2}

Area of the base of the hemisphere = πr^{2}, 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 l^{2} + 2πr^{2} - πr^{2}

= 6 l^{2} + πr^{2}

= 6 l^{2} + π (l/2)^{2}

= 6 l^{2} + πl^{2} / 4

= ¼ l^{2} (π + 24)

Thus, the surface area of the remaining solid is ¼ l^{2} (π + 24).

**Video Solution:**

## 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

### NCERT Solutions Class 10 Maths - Chapter 13 Exercise 13.1 Question 5 :

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

The surface area of the remaining solid if 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 is ¼ l^{2} (π + 24)