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.

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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

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:

-    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.

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.

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.