Surface Area and Volumes Class 10 Notes Maths Chapter 12

Formulas for Surface Area and Volume of various Solids

Surface Area and Volume of Combinations

1. Cone on a Cylinder

Slant height, l = 

2. Cone on a Hemisphere

h = height of cone; 
I = slant height of cone =
r = radius of cone and hemisphere 
Total Surface area = Curved surface area of cone + Curved surface area of hemisphere = πrl + 2πr² 
Volume = Volume of cone + Volume of hemisphere =

3. Conical Cavity in a Cylinder

r = radius of cone and cylinder;
h = height of cylinder and conical cavity;
l = Slant height
Total Surface area = Curved surface area of cylinder + Area of the bottom face of cylinder + Curved surface area of cone = 2πrh + πr² + πrl
Volume = Volume of cylinder – Volume of cone = 

4. Cones on Either Side of the Cylinder

r = radius of cylinder and cone;
h₁ = height of the cylinder
h₂ = height of cones
Slant height of cone, l = 
Surface area = Curved surface area of 2 cones + Curved surface area of cylinder = 2πrl + 2πrh₁ 
Volume = 2(Volume of cone) + Volume of cylinder = 

5. Cylinder with Hemispherical Ends

r = radius of cylinder and hemispherical ends;
h = height of cylinder
Total surface area= Curved surface area of cylinder + Curved surface area of 2 hemispheres = 2πrh + 4πr²
Volume = Volume of cylinder + Volume of 2 hemispheres =  πr²h+4/3πr³

6. Hemisphere on Cube or Hemispherical Cavity on Cube

a = side of cube;

r = radius of hemisphere.

Surface area = Surface area of cube – Area of hemisphere face + Curved surface area of hemisphere

= 6a² – πr² + 2πr² = 6a² + πr²

Volume = Volume of cube + Volume of hemisphere =

7. Hemispherical Cavity in a Cylinder

r = radius of hemisphere;
h = height of cylinder
Total surface area = Curved surface area of cylinder + Surface area of base + Curved surface area of hemisphere
= 2πrh + πr² + 2πr² = 2πrh + 3πr²
Volume = Volume of cylinder – Volume of hemisphere =

Some Solved Examples

Example 1: A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Sol:

The figure drawn below of the vessel and lead shot is to visualize it.

As the water is filled up to the brim in the vessel

Volume of water in the vessel = Volume of the conical vessel

On dropping a certain number of lead shots (sphere) into the vessel one-fourth of the water flows out.

Volume of all lead shots dropped into the vessel = 1/4 × Volume of the water in the vessel

Hence,

Number of lead shots = 1/4 × volume of the water in the vessel ÷ volume of each lead shot

We will find the volume of the water in the vessel and lead shot by using formulae;

Volume of the sphere = 4/3 πr³

where r is the radius of the sphere

Volume of the cone = 1/3 πR²h

where R and h are the radius and height of the cone respectively

Height of the conical vessel, h = 8 cm

Radius of the conical vessel, R = 5 cm

Radius of the spherical lead shot, r = 0.5 cm

Number of lead shots = 1/4 × volume of the water in the vessel ÷ volume of each lead shot

= 1 /4 × (1/3) πR²h × 3/4 πr³

= πR²h/12 × 3/4πr³

= R²h / 16r³

= (5cm × 5 cm × 8 cm) / (16 × 0.5 cm × 0.5 cm × 0.5 cm)

= 100

Thus, the number of lead shots dropped in the vessel is 100.

Example 2: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.

Sol:

The visualization of the solid figure is drawn below.

The volume of a solid is the space occupied inside it or the capacity that an object holds.

As the solid is made up of a conical part and a hemispherical part,

Volume of the solid = volume of the conical part + volume of the hemispherical part

Let us find the volume of the solid by using formulae;

Volume of the hemisphere = 2/3 πr3 where r is the radius of the hemisphere

Volume of the cone = 1/3 πr2h where r and h are the radius and height of the cone respectively.

Radius of hemispherical part = Radius of conical part = r = 1 cm

Height of conical part = h = r = 1 cm

Volume of the solid = volume of the conical part + volume of the hemispherical part

= 1/3 πr²h + 2/3 πr³

= 1/3 πr³ + 2/3 πr³ [Since, h = r]

= πr³

= π (1cm)³

= π cm³

Thus, the volume of the solid is π cm³.