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?
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
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
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.