Cone Formulas
Slant Height (l) = √(r² + h²)
Lateral Surface Area = π × r × l
Total Surface Area = π × r × (r + l)
Where r = base radius, h = height, l = slant height, π ≈ 3.14159...
Example Calculation
Cone with radius = 4 cm, height = 9 cm
Slant height = √(4² + 9²) = √(16 + 81) = √97 ≈ 9.849 cm
Volume = (1/3) × π × 16 × 9 = 150.796 cm³
Lateral SA = π × 4 × 9.849 = 123.772 cm²
Total SA = π × 4 × (4 + 9.849) = 174.027 cm²
Key Facts About Cones
- A cone's volume is exactly 1/3 of a cylinder with the same base and height.
- The slant height is the distance from the apex to the edge of the base circle, calculated via the Pythagorean theorem.
- An ice cream cone with radius 3 cm and height 12 cm holds about 113 cm³ (113 mL).
- A traffic cone with radius 15 cm and height 75 cm has a volume of about 17,671 cm³ (17.7 litres).
Frequently Asked Questions
Why is the cone volume 1/3 of a cylinder?
This can be proven by calculus (integration) or physically demonstrated by filling a cone with water and pouring it into a cylinder of the same base and height — it takes exactly three cone-fuls to fill the cylinder. The factor of 1/3 comes from the way the cross-section shrinks linearly toward the apex.
What is slant height and how is it different from height?
Height (h) is the perpendicular distance from the base to the apex — measured straight up. Slant height (l) is the distance along the surface from the apex to the base edge — it's always longer than the height. They relate by: l = √(r² + h²).
How do I find the volume if I know the slant height instead of the height?
Rearrange the slant height formula: h = √(l² − r²). Then use V = (1/3)πr²h as usual.
Can I calculate a frustum (truncated cone)?
A frustum is a cone with the top cut off. Its volume is V = (π×h/3)×(R² + R×r + r²), where R is the large radius and r is the small radius. This calculator handles full cones only.