# CLASS-10VOLUME & SURFACE AREA OF THE SOLID - SPHERE

SPHERE

The word name “Sphere” is usually comes in our mind as circle or round shape object. As an example the objects such as football, volleyball, throw-ball, etc., are said to have the shape of a sphere. When a circular lamina is revolved about any of its diameters, then the solid generated is called a “SPHERE”. The center and radius of this circle are respectively the center and radius of the sphere.

Sphere -

Formulae -

1. For a solid sphere of radius = r, we have –

4

(i) Volume of the sphere = (------ πr³) cubic units

3

(ii) Surface area of the sphere =  (4πr²)  sq. Units

Spherical Shell

The solid enclosed between two concentric sphere is called a spherical shell.

2. For a spherical shell with external radius = R, and internal radius = r, we hve –

(i) Thickness of shell = (R – r) units,

4

(ii) Volume of the material = ------ π(R³ - r³) cubic units

3

Hemisphere

When a plane through the center of a sphere cuts it into two equal parts, then each part is called a hemisphere.

3. For a Hemisphere of Radius r, we have –

2

(i)  Volume = ------ πr³ cubic units

3

(ii) Curved Surface Area = 2πr² sq. Units

(iii)  Total Surface area = (2πr²+ πr²) = 3πr² sq. Units

There are some examples are given below for your better understanding -

Example.1)  Find the volume and surface area of a sphere whose diameter is 42 cm. (Take π = 22/7)

42

Ans.)  Radius of the sphere, r = -------- =  21 cm

2

4

So, volume of the sphere = ----- πr³

3

4           22

= ------- X ------- X 21 X 21 X 21

3            7

=  (4 X 22 X 21 X 21) = 38808 cm³  (Ans.)

Example.2)  The volume of a sphere is 36π cm³. Find the surface area of the sphere, correct to nearest cm² (Take π = 22/7)

Ans.)  Let the radius of the sphere be r cm.

4

Then, its volume = (------ πr³) cm³

3

As per the given condition –

4

--------- πr³  =  36π

3

=>           4r³ =  (36 X 3)

=>            r³ =  (9 X 3) = 3³

=>             r = 3

So, radius of the sphere is 3 cm

Now, the surface area of the sphere = 4πr²

22

=  4 X ------- X 3 X 3

7

792

= -------- =  113.14 cm²

7

Hence the surface area of the sphere, correct to nearest cm² is 113 cm²    (Ans.)

Example.3)  A hollow sphere of internal and external radii 9 cm and 12 cm respectively is melted and recast into small cones of base radius 2 cm and height 6 cm. Find the number of cones formed.

Ans.)  External radius of the sphere (R) = 12 cm

Internal radius of the sphere (r) = 9 cm

So, volume of metal obtained from the hollow sphere

4

= ----- π (R³ - r³)

3

4          22

= ------ X ------ X (12³ - 9³)

3           7

88

= ------ X (1728 – 729)

21

88 X 999      88 X 333

= ---------- = ---------- = 1332π

21               7

Radius of each cone = 2 cm, and its height = 6 cm

1                  1

Volume of each cone = (------ πr²h) = (------ π X 2² X 6)

3                  3

=   8 π cm³

Volume of Metals

Number of cones formed =  ----------------------

Volume of each cone

1332 π

=  -----------  = 166.5 = 166

8π

Hence, 166 cones are formed.     (Ans.)

Your second block of text...