Surface Area of a Sphere

If we cover half of the surface of a sphere with a cord as shown in the figure, and then wind the exact same length of cord around the surface of a cylinder which has the same radius and a height 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 $= 2r$, with $r$ 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.

Summary:

1. The area of the surface of a sphere $= 4\pi {r^2}$ or $\pi {d^2}$ sq.units, where$r$ is the radius and $d$ is the diameter.
2. The total surface area of the hemisphere $= 3\pi {r^2}$ sq.units

Example:

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

Solution: $= \frac{{36}}{2} = 18$cm
Length of the cylinder $= \left( {108 – 18 – 18} \right) = 72$cm
Surface of the cylinder $= 2\pi rh$
Surface of the two hemispheres $= 2 \times \left( {\frac{1}{2}4\pi {r^2}} \right)$
Total surface $= 2\pi rh + 4\pi {r^2} = 2\pi r\left( {h + 2\pi } \right)$
$= 2 \times 3.14 \times 18\left( {72 + 2 \times 18} \right)$
$= 12214.54$sq.cm (approx)

Cost of polishing at 7 Dollars per sq.cm $= \frac{7}{{100}} \times 12214.54 = 855.02$

Example:

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.

Solution: Surface area of hemisphere $= \frac{1}{2}\left( {4\pi {r^2}} \right) = 2 \times 3.14 \times {3^2}$
$= 56.25$ sq.cm
Now, for the conical part, we have
$r = 3$cm, $h = 4$cm
$\therefore$            $l = \sqrt {{r^2} + {h^2}} = \sqrt {9 + 16} = 5$cm

Also, the curved surface of the cone $= \pi rl = 3.14 \times 3 \times 5 = 47.10$cm
$\therefore$            Surface area of the toy $= 56.52 + 17.10 = 103.62$ sq.cm