Volume of Part of a Sphere

Definition of Hemisphere
A plane through the centre of  a sphere divides it into two equal parts. Each part is called the hemisphere.
Let 'r' be the radius of a hemisphere.Then,
Volume of the hemisphere is  2/3πr3  cubic units.That means Volume of half the part of a sphere is 2/3πr3 cubic units.
                                                                                                                                                                          
Curved surface area of the hemisphere = 2 π r2 square units.
Total surface area of the hemisphere is 3πr2 square units.
                                                                        

Problems on Volume of Part of a Sphere

 

Let us do some problems on volume of part of a sphere.

#Find the volume of a hemisphere  of radius 7 cm.
Solution:- Volume of the sphere is 2/3πr3 cu. units =  2/3(3.14) (7)3 =  718.01 cu. cm
                                                                                                
# Find the surface area of a hemisphere of radius 7 cm.
Solution : Surface area = 2 π r2 =  2 (3.14) ( 7)2 =   307.72 sq. cm
 
#The circumference of  the edge of a hemispherical bowl is 132 cm. What is its capacity?
Solution:-
Let r be the radius of the hemisphere.
Then circumference of the edge is 2πr = 132 cm
Hence radius = 132 / 2π = 132 ÷ (2x3.14) = 132 ÷ 6.28 = 21.36 cm
Capacity of the sphere =Volume of the hemisphere = 2/3πr3 = ( 2 x 3.14 x 21.36 x21.36 x 21.36) ÷ 3  = 20,400.56 cu cm.                                    

Volume of Part of a Sphere-hollow Hemisphere:-

 

 A hollow hemisphere contains a big hemisphere outside and a small hemisphere inside.(example just think of a coconut broken in two equal parts.)  Its external radius  is 'R' and internal radius is 'r'.
Volume of the hemisphere is 2/3π(R3  - r3 )  cubic units.
                                            
# A hemispherical hollow bowl has material of volume 436π cu.cm.  Its external diameter is 14 cm.  Find its thickness.                                                                          3
                                                                                                       
Let R be the external radius and r be the internal radius,
External radius R = 14/2 = 7 cm
Volume of the material  which is volume of half the part of the sphere is= 2/3 π(R3 - r3)   = 436π
                                                                                                                                           3                            
Cancelling π/3 on both sides we get  2(R3 - r3)  = 436 =>  R3 - r3 = 218 => 73 - r 3 = 218 => 343 - r3 = 218 =>
                                  -r3 = -125=> r3 = 53
Thus we get  r = 5 cm
Thickness = R - r = 7 - 5 = 2 cm
Thus by using volume of part of a sphere , we have calculated the thickness of hemispherical bowl.