1. Understanding a Sphere
A sphere is a perfectly round 3D object where every point on the surface is the same distance from the centre. This distance is called the radius \(r\).
Examples include balls, marbles, bubbles, and planets. The volume of a sphere tells us how much space is enclosed inside it.
2. Formula for Volume of a Sphere
Unlike cubes or cylinders, the volume of a sphere cannot be derived using simple length × width × height ideas. Instead, it comes from geometry involving circular cross-sections.
The formula for the volume of a sphere of radius \(r\) is:
\( \text{Volume} = \dfrac{4}{3} \pi r^3 \)
This formula shows that the volume is proportional to the cube of the radius.
2.1. Example
Find the volume of a sphere with radius \(r = 7\,\text{cm}\).
\( \text{Volume} = \dfrac{4}{3} \pi r^3 = \dfrac{4}{3} \pi \times 343 \)
\( = \dfrac{4 \times 343}{3} \pi = \dfrac{1372}{3} \pi \approx 457.33\pi \)
Using \( \pi = 3.14 \):
\( 457.33 \times 3.14 = 1436.02\,\text{cm}^3 \)
3. Why the Formula Makes Sense (Simple Intuition)
A sphere can be seen as being made of many circular slices stacked from top to bottom. Each slice has a different radius, and integrating these areas gives the formula involving \(r^3\).
Even without calculus, it's important to remember: Increasing the radius slightly increases the volume a lot, because it grows with the cube of the radius.
4. Real-Life Applications
The volume of a sphere is useful in:
- Designing sports balls (football, table tennis ball)
- Measuring capacity of spherical tanks
- Calculating air inside balloons
- Astronomy—estimating the size of planets or stars
Whenever the object is completely round, use \(\dfrac{4}{3} \pi r^3\) to find its volume.