1. Understanding a Cone
A right circular cone is a 3D solid with:
- a circular base of radius \(r\)
- a vertex (a sharp point)
- a height \(h\), measured perpendicular to the base
Common examples include ice-cream cones, funnel shapes, conical tents, and traffic cones. The volume tells us how much space the cone can hold inside.
2. Formula for Volume of a Cone
The cone's volume is exactly one-third of the volume of a cylinder with the same base and height.
This happens because cones taper to a point, while cylinders have uniform cross-sections.
The formula is:
\( \text{Volume} = \dfrac{1}{3} \pi r^2 h \)
2.1. Example
A cone has radius \(r = 6\,\text{cm}\) and height \(h = 12\,\text{cm}\).
\( \text{Volume} = \dfrac{1}{3} \pi r^2 h = \dfrac{1}{3} \pi \times 36 \times 12 \)
\( = \dfrac{1}{3} \times 432\pi = 144\pi \)
Using \(\pi = 3.14\):
\( 144 \times 3.14 = 452.16\,\text{cm}^3 \)
3. Why the Volume Formula Works
If you fill a cone completely with water and pour it into a cylinder with the same radius and height, you need exactly three full cones to fill the cylinder.
This explains why the cone’s volume is one-third of the cylinder’s volume:
\( \text{Volume of cone} = \dfrac{1}{3} \pi r^2 h \)
4. Real-Life Applications
Volume of cones is often used in:
- Finding the capacity of funnel-shaped containers
- Designing ice-cream cones
- Measuring sand or grain stored in conical heaps
- Calculating material needed to form conical solid objects
Whenever a shape tapers to a point, use the formula \(\dfrac{1}{3} \pi r^2 h\) to find the volume.