Surface Area of a Sphere

If we wind half of the surface of sphere with cord as shown in the figure, and then wind with exactly the same length of the cord the surface of a cylinder having just the same radius, the height of the cylinder equaling the diameter, we find that the cord covers half of the curved surface of the cylinder.


Therefore, the surface area of a sphere equals the curved surface area of a cylinder of the same radius and height.

Here the height of the cylinder  = 2r, r being the radius of the sphere and also the radius of the circular base of the cylinder.
            \therefore             Area of the curved surface of the cylinder  = 2r \times 2\pi r = 4\pi {r^2}
                        Hence, the surface of a sphere, S = 4\pi {r^2}
Note, that the surface of a sphere is equal to the area of four of its great circles.


  1. Area of the surface of a sphere  = 4\pi {r^2} or \pi {d^2} sq.units, whereris the radius anddis the diameter.
  2. Total surface area of the hemi-sphere  = 3\pi {r^2} sq.units


A solid is composed of a cylinder with hemisphere ends. If the whole length of the solid is 108cm and diameter of the hemi-spherical ends is 36cm, find the cost of polishing the surface at the rate of 7 Dollars per



            Radius of the cylinder = Radius of the hemi-sphere
                                                 = \frac{{36}}{2} = 18cm
            Length of the cylinder  = \left( {108 - 18 - 18}  \right) = 72cm
            Surface of the cylinder  = 2\pi rh
            Surface of two hemispheres  = 2 \times \left( {\frac{1}{2}4\pi  {r^2}} \right)
            Total surface  = 2\pi rh + 4\pi {r^2} = 2\pi  r\left( {h + 2\pi } \right)
                                                                = 2  \times 3.14 \times 18\left( {72 + 2 \times 18} \right)
                                                                = (approx)
            Cost of polishing at 7 Dollars per  = \frac{7}{{100}} \times 12214.54 = 855.02


A toy is in the form of a cone mounted on a hemisphere. The diameter of the base of the cone is 6cm and its height is 4cm. Calculate the surface area of the toy.



            Surface area of hemisphere  = \frac{1}{2}\left( {4\pi {r^2}} \right) = 2  \times 3.14 \times {3^2}
                                                          =  56.25
            Now, for the conical part, we have
                        r = 3cm, h  = 4cm
            \therefore             l = \sqrt {{r^2} +  {h^2}} = \sqrt {9 + 16} = 5cm
            Also, the curved surface of the cone  = \pi rl = 3.14  \times 3 \times 5 = 47.10cm
            \therefore             Surface area of the toy   = 56.52 + 17.10 = 103.62