1. Understanding a Hemisphere
A hemisphere is exactly half of a sphere. It has:
- a curved surface
- a flat circular base
The radius of the hemisphere is the same as the radius of the sphere from which it is formed. The volume of a hemisphere tells us how much space is inside it — similar to measuring how much water it can hold.
2. Formula for Volume of a Hemisphere
Since a hemisphere is half of a sphere, its volume is half of the sphere's volume.
The volume of a sphere is:
\( \dfrac{4}{3} \pi r^3 \)
So, the volume of a hemisphere is:
\( \text{Volume of hemisphere} = \dfrac{1}{2} \times \dfrac{4}{3} \pi r^3 = \dfrac{2}{3} \pi r^3 \)
2.1. Example
A hemisphere has radius \(r = 6\,\text{cm}\). Its volume is:
\( \dfrac{2}{3} \pi r^3 = \dfrac{2}{3} \pi (6^3) = \dfrac{2}{3} \pi (216) = 144\pi \)
Using \(\pi = 3.14\):
\( 144 \times 3.14 = 452.16\,\text{cm}^3 \)
This means the hemisphere can hold about 452 cubic centimetres of space.
3. Understanding the Formula (Simple Intuition)
Imagine slicing the sphere into two equal halves. Each half contains exactly half the space.
So instead of using \(\dfrac{4}{3} \pi r^3\), we simply take half of it:
\( \dfrac{4}{3} \pi r^3 \div 2 = \dfrac{2}{3} \pi r^3 \)
Since the hemisphere has a flat base, but we are calculating volume, the base does not change the formula.
4. Real-Life Applications
Volume of hemispheres is used in many real-world objects and measurements, such as:
- Bowls and containers shaped like hemispheres
- Dome-shaped structures
- Liquid capacity of half-spherical tanks
- Calculating volume of ladles, cups, and lampshades
Whenever the object is shaped like half of a sphere, use \(\dfrac{2}{3} \pi r^3\) to find its internal capacity.