Cone Calculator
Enter the radius and height to get a cone’s volume, slant height, and lateral and total surface area.
Slant height l = 5 · V = ⅓πr²h = ⅓·π·3²·4
A right circular cone with radius r = 3 and height h = 4 has slant height l = √(3² + 4²) = 5, volume V = ⅓π×3²×4 ≈ 37.70 cubic units, and total surface area πr(r + l) = π×3×8 ≈ 75.40 square units. Its lateral (side) area alone is π×3×5 ≈ 47.12.
What the cone calculator finds
A right circular cone is a circle of radius r joined to a single apex a perpendicular height h above the centre. From those two numbers everything else follows: the slant height l (apex to the rim along the surface), the volume, and two areas — the lateral surface (the curved side) and the total surface (side plus the circular base).
volume, slant height (Pythagoras), and total surface area
Worked example
Take a cone with radius r = 3 and height h = 4 — a 3-4-5 right triangle in cross-section:
- 1 Find the slant height. l = √(r² + h²) = √(3² + 4²) = √(9 + 16) = √25 = 5. The radius, height, and slant form a right triangle.
- 2 Compute the volume. V = ⅓πr²h = ⅓ × π × 3² × 4 = 12π ≈ 37.70 cubic units.
- 3 Find the base and lateral areas. Base = πr² = 9π ≈ 28.27. Lateral = πrl = π × 3 × 5 = 15π ≈ 47.12 square units.
- 4 Add for the total surface area. SA = πr(r + l) = π × 3 × (3 + 5) = 24π ≈ 75.40 square units — the base plus the curved side.
Cone measures for common r and h
Volume in cubic units, slant and total surface area computed from V = ⅓πr²h, l = √(r² + h²), SA = πr(r + l).
| Radius r | Height h | Slant l | Volume V | Total SA |
|---|---|---|---|---|
| 3 | 4 | 5 | 37.70 | 75.40 |
| 5 | 12 | 13 | 314.16 | 282.74 |
| 6 | 8 | 10 | 301.59 | 301.59 |
| 1 | 1 | 1.4142 | 1.05 | 7.58 |
| 2 | 2 | 2.8284 | 8.38 | 30.34 |
Why the volume is a third of the cylinder
A cone sitting inside a cylinder of the same radius and height fills exactly one third of it — that is the ⅓ in V = ⅓πr²h. Because a cone tapers to a point while a cylinder has a full circle at every level, integrating the shrinking cross-sections gives one third of the πr²h a cylinder would hold. The same one-third factor applies to any pyramid versus its prism.
Slant height is not the vertical height. The height h runs straight up from the centre of the base to the apex; the slant height l runs along the surface from the rim to the apex. They differ by Pythagoras — l = √(r² + h²) is always the longest — so use l (never h) for the curved surface area, and h (never l) for the volume.