Sphere

Rotate a semicircle about its diameter. The solid so generated with this rotation is called a sphere.

The locus of a point which moves in space in such a way that its distance from a fixed point remains the same is called a sphere. The fixed point is called the centre of the sphere and the same distance is called the radius of the sphere. A football, cricket ball, are examples of spheres.

Hemisphere

If a sphere is cut into two equal parts by a plane passing through its centre, then each part is called a hemisphere.

Surface Areas

Take a spherical rubber (or wooden) ball and cut it into equal parts (hemisphere). Let the radius of the ball be r. Now, put a pin (or a nail) at the top of the ball. Starting from this point, wrap a string in a spiral form till the upper hemisphere is completely covered with string. Measure the length of the string used in covering the hemisphere.

Now draw a circle of radius r (the same radius as that of the ball and cover it with a similar string starting from the centre of the circle. Measure the length of the string used to cover the circle. You will observe that length of the string used to cover the hemisphere is twice the length of the string used to cover the circle.

Since the width of the two strings is the same, therefore

surface area of the hemisphere = 2 × area of the circle

= 2 πr²

So, surface area of the sphere = 2 × 2πr² = 4πr²

Surface area of a sphere = 4πr²

Curved surface area of a solid hemisphere = 2πr² + πr² = 3πr²

Volumes

Take a hollow hemisphere and a hollow right circular cone of the same base radius and same height (say r). Now fill the cone with sand (or water) and pour it into the hemisphere. Repeat the process two times. You will observe that hemisphere is completely filled with the sand (or water). It shows that volume of a hemisphere of radius r is twice the volume of the cone with same base radius and same height.

$$ \text{Volume of hemisphere} = 2 \times \frac{1}{3} \pi r^2 $$

As h = r

$$ \text{Volume of hemisphere} = \frac{2}{3} \pi r^3 $$

$$ \text{Volume of sphere} = \frac{4}{3} \pi r^3 $$