Hemisphere

Hemisphere

Hemisphere

Hemisphere is half of a sphere. When a sphere is divided equally into two parts, each part is called hemisphere. It has two types of surfaces, one is circular face (great circle) and another is curved surface. The radius of hemisphere is equal to the radius of sphere.
Sphere and hemisphere

Total Surface Area of a Hemisphere

Total surface area of a hemisphere is the sum of its area of curved surface and the area of circular face (great circle). Area of curved surface of a hemisphere is the half of the total surface area of a sphere.

So, area of curved surface of a hemisphere = ½ of total surface area of the sphere
                                                                      = ½ × 4πr2
                                                                      = 2πr2

And, area of circular face (great circle) = πr2
Total Surface Area a of Hemisphere
Now,
Total surface area of hemisphere = area of curved surface + area of circular face
                                                      = 2πr2 + πr2
                                                      = 3πr2

   Total surface of hemisphere = 3πr2


Volume of a Hemisphere

To find the volume of a hemisphere, we can do the following activity:
Activity to find the volume of hemisphere
-    Take a cylindrical vessel circumscribing a plastic spherical ball as given in the figure above.
-    Take out the spherical ball from the cylinder and cut it into two equal parts. In this way we will get two hemispheres of plastic ball.
-    Fill a hemisphere with rice or sand and pour it into the cylinder. And repeat it again.
-    Then we will find that the cylinder will be full filled by 3 hemispheres.

Thus, the volume of 3 hemisphere = volume of 1 cylinder  
                                                        = πr2h    [ volume of cylinder = πr2h]
                                                        = πr2 × d    [ h = d]
                                                        = πr2 × 2r    [ d = 2r]
                                                        = 2πr3
             Volume of 1 hemisphere = 2πr3/3

Thus, the volume of a hemisphere = 2πr3/3


Workout Examples

Example 1: Find the total surface area and the volume of a hemisphere of radius 3.5cm.
Example 1: Hemisphere

Solution: Here, radius of hemisphere (r) = 3.5cm
                Now,
                         Total surface area of hemisphere = 3πr2
                                                                                  = 3 × 22/7 × (3.5)2
                                                                                  = 115.50cm2

                         Volume of hemisphere = 2πr3/3
                                                                 = 2/3 × 22/7 × (3.5)3
                                                                 = 89.83cm3

                Thus, total surface area is 115.50cm2 and the volume is 89.83cm3.


Example 2: The circumference of the edge of a hemispherical bowl is 132cm. Find the capacity of the bowl.

Solution: Here, circumference = 132cm
                i.e.      2πr = 132
                or,      2 × 22/7 × r = 132
                or,      r = 132 × 7/44
                or,      r = 21cm

                Now,
                          The capacity of the bowl = 2πr3/3
                                                                     = 2/3 × 22/7 × (21)3
                                                                     = 19404cm3  
         
                Thus, the capacity of the bawl is 19404cm3.


Example 3: Find the total surface area and the volume of the given combined solid figure.
Example 3: Combined solid

Solution: Here, the given combined solid figure is a cylinder and a hemisphere,
    radius (r) = 7cm
    height (h) = 10cm

                Now,
                         Total surface area of solid = 2πr2 + 2πrh + πr2
                                                                       = 3πr2 + 2πrh
                                                                       = 3 × 22/7 × (7)2 + 2 × 22/7 × 7 × 10
                                                                       = 462 + 440
                                                                       = 902cm2

                         Volume of solid = πr2h + 2πr3/3
                                                     = 22/7 × 72 × 10 + 2/3 × 22/7 × 73
                                                     = 1540 + 718.67
                                                     = 2258.67cm3

                Thus, total surface area is 902cm2 and the volume is 2258.67cm3.


You can comment your questions or problems regarding the surface area and volume of hemisphere here.

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