CLASS-10
VOLUME & 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.)
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