Sphere and Hemisphere

We are so much familiar with spherical objects. The sphere is also a solid object whose each point in the outer surface is equidistance from the fixed point inside it. Such a fixed point is called the centre of the sphere. The constant distance is called the radius of the sphere. A solid object such as a globe, volleyball, toy ball, table tennis ball, marble etc is the example of a sphere.

The figure as shown alongside is a sphere. The fixed point 'O' inside it is the centre which is equidistance from each point P on the surface of the sphere. So, OP = r is the radius of the sphere.

If we cut a sphere through its diameter, there are two half spheres called the hemisphere and the cross section is called the great circle. The radius of the sphere is same as the radius of the great circle. The centre of the sphere and its great circle. The centre of the sphere and its great circle is same.

Surface area of sphere

The surface area of a sphere is the area of its outer part, which is a smooth curved surface.

The surface area of a sphere is given by SA = 4πr² where r is the radius of the sphere. The total surface area of hemisphere = 2πr²+ πr² = 3πr² square unit.

Volume of sphere

The volume of a sphere means the space that it occupies. We can measure the volume of sphere experimentally. Fill up the measuring cylinder with the water level in the cylinder. The difference of two levels is the volume of the sphere.

Alternatively

Choose a sphere of given diameter (d) =10 cm (say).
Immerse of the whole sphere into the water in the measuring cylinder, the water level is raised by 523.33 ml. Therefore its volume is 523.33 cm³. From this experiment,
The diameter (d) = 10cm
The volume of sphere (V) = 523.33 cm³
Let us make the ratio

\begin{align*} 6V:d^3 &= \frac {6V} {d^3} \\ &= \frac {6 \times 523.33} {10^3} \\ &= \frac {3140} {1000} \\ &= 3.14 \end{align*}

$$ (\because \pi = \frac {22} {7} = 3.14) $$

\begin{align*} \therefore \frac {6V} {d^3} &= \pi \\ or, V &= \frac {\pi (2r)^3} {6} \\ &= \frac {4 \pi r^3} {3} \\ \end{align*}

\( \therefore \text {Volume of a sphere} (V) = \frac {4 \pi r^3} {3} = \frac {\pi d^3} {6} \text {cubic units.} \)