Cone Formulas
A cone is a three-dimensional shape with a circular base that tapers smoothly to a single point called the apex. The distance from the center of the base to the apex is the height (h), and the circular base has radius (r).
Slant Height l = √(r² + h²)
Lateral Surface Area = π × r × l
Base Area = π × r²
Total Surface Area = πrl + πr² = πr(l + r)
The 1/3 Factor Explained
A cone's volume is exactly one-third of the cylinder with the same radius and height. You can visualize this: it takes exactly 3 cones filled with water to fill one cylinder of identical base and height.
Ratio: Cone : Cylinder = 1 : 3
Similarly, a pyramid has 1/3 the volume of a prism with the same base and height — the 1/3 rule applies to all pointy 3D shapes.
Worked Examples
Example 1 – Ice Cream Cone
An ice cream cone has radius 3 cm and height 12 cm. Find its volume.
Example 2 – Traffic Cone
A traffic cone has radius 15 cm and height 70 cm. Find slant height and surface area.
Lateral SA = π × 15 × 71.59 ≈ 3374.8 cm²
Total SA = 3374.8 + π × 225 ≈ 4081.5 cm²
Example 3 – Sand Pile
A conical sand pile has base diameter 6 m and height 2 m. How many cubic metres of sand?
V = (1/3) × π × 9 × 2 = 6π ≈ 18.85 m³ of sand
Comparison: Cone, Cylinder, Sphere (same r)
| Shape | Volume Formula | r=3, h=6 |
|---|---|---|
| Cylinder | πr²h | 169.65 |
| Sphere (r=3) | (4/3)πr³ | 113.10 |
| Cone | (1/3)πr²h | 56.55 |
Real-World Applications
- Food: Ice cream cones, party hats, conical paper cups
- Traffic safety: Traffic cones — engineers need to calculate material for manufacturing
- Construction: Conical rooftops (spires), funnel-shaped hoppers for grain or cement
- Geology: Volcanic cones, sand dunes, scree slopes
- Optics: Conical shapes in lenses and optical instruments