Surface Area of a Sphere

Surface Area of a Sphere

The total curved surface around the sphere is called the total surface area (TSA) of the sphere. Archimedes (6^th century BC) discovered that the total surface area of a spherical object like tennis ball, football etc. is equal to the curved surface area of the circumscribing cylinder.

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The figure given above is the cylinder circumscribing a sphere. Here diameter of the sphere AB is equal to the diameter PQ of the cylinder and diameter of sphere is equal to the height QR of cylinder.

We know that the curved surface area (CSA) of a cylinder = 2πrh

Now, by Archimedes principle on TSA of sphere,

Total surface area of sphere = curved surface area of the circumscribing cylinder

                                              = 2πrh

                                              = 2πr × d      [∵ h = d]

                                              = 2πr × 2r    [∵ d = 2r]

                                              = 4πr²

∴  Total surface area of a sphere = 4πr²

Surface area of a hemisphere

Hemisphere is one of the half part of a sphere when it is divided into two equal parts. It has two surfaces, one circular face (great circle) and another curved surface.

The radius of hemisphere is equal to the radius of the sphere. Since hemisphere is half part of sphere,

Curved surface area of hemisphere = ½ of curved surface area of sphere

                                                          = ½ × 4πr²

                                                          = 2πr²

Total surface area of hemisphere = CSA of hemisphere + area of circular surface

                                                      = 2πr² + πr²

                                                      = 3πr²

∴  TSA of hemisphere = 3πr²

Workout Examples

Example 1: Find the total surface area of a sphere of diameter 21cm.

Solution: Here,

                          Diameter of sphere (d) = 21cm

                          ∴  radius of sphere (r) = 21/2 = 10.5cm

                Now,

                          TSA of sphere = 4πr²

                                                   = 4 × 22/7 × (10.5)²

                                                   = 88/7 × 110.25

                                                   = 1386cm²

                Thus, the total surface area of sphere is 1386cm².

Example 2: Find the total surface area of a sphere if the circumference of the great circle is π cm.

Solution: Here,

                          Circumference of great circle = π cm

                i.e.     2πr = π

                or,     r = π/2π

                or,     r = ½

                or,     r = 0.5cm

                Now,

                          TSA of sphere = 4πr²

                                                    = 4 × π × (0.5)²

                                                    = 4 × π × 0.25

                                                    = π cm²

                Thus, the total surface area of sphere is π cm².

Example 3: Find the total surface area of two hemispheres when a sphere of radius 7cm is cut into two halves.

Solution: Here,

                          Radius of hemisphere = radius of sphere = 7cm

                Now,

                          The total surface area (TSA) of a hemisphere = 3πr²

                                                                                                        = 2 × 22/7 × 7²

                                                                                                        = 2 × 22 × 7

                                                                                                        = 308cm²

                ∴  Total surface area of two hemisphere = 2 × 308cm²

                                                                                     = 616 cm²

                Thus, the total surface area of two hemisphere is 616 cm².

You can comment your questions or problems regarding the TSA of sphere and hemisphere here.