# A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

**Solution:**

We can create the following figure of the solid as per the given information.

From the figure, it’s clear that the greatest diameter the hemisphere can have is equal to the cube’s edge. Otherwise, a part of the hemisphere’s rim (circumference of its circular base) will lie outside the top part of the cube.

As it’s clear from the top view of the figure that the top part of the cube is partially covered by the hemispherical region.

Total surface area of the solid = Surface area of the cubical part - Area of the base of the hemispherical part + CSA of the hemispherical part

We will find the total area of the solid by using the formulae;

CSA of the hemisphere = 2πr^{2}

Area of the base of the hemisphere = πr^{2}, where r is the radius of the hemisphere

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

Length of the edge of the cube, l = 7 cm

From the figure, it’s clear that the greatest diameter the hemisphere can have is equal to the cube’s edge

Diameter of the hemisphere, d = l = 7 cm

Radius of the hemisphere, r = d/2 = 7/2 cm

Total surface area of the solid = Surface area of the cubical part - Area of the base of the hemispherical part + CSA of the hemispherical part

= 6l^{2} - πr^{2} + 2πr^{2}

= 6l^{2} + πr^{2}

= 6 × (7 cm)^{2} + 22/7 × (7/2 cm)^{2}

= 6 × 49 cm^{2} + 22/7 × 49/4 cm^{2}

= 294 cm^{2} + 38.5 cm^{2}

= 332.5 cm^{2}

Thus, the greatest diameter the hemisphere can have is 7 cm and the surface area of the solid is 332.5 cm^{2}.

**Video Solution:**

## A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

### NCERT Solutions Class 10 Maths - Chapter 13 Exercise 13.1 Question 4:

**Summary:**

The greatest diameter of a hemisphere which surmounts a cubical block of side 7 cm is 7 cm and the surface area of the solid is 332.5 cm^{2}.