Updated · Built and checked by the PlinzoTools team, formulas tested against published sources
Calculate a cone's volume, surface area, and slant height from its radius and height.
This tool calculates a right circular cone's volume, total surface area, and slant height from its base radius and height.
Enter the radius and height. Volume, surface area, and slant height all appear instantly.
Slant Height = √(r² + h²)
Volume = (1/3) × πr²h
Surface Area = πr(r + slant height)
This calculator returns three numbers, and each answers a different question. Slant height is a length: the straight-line distance along the outside of the cone from the rim of the base up to the tip. Volume is a capacity in cubic units. Surface area is the total outside covering in square units, counting both the curved side and the flat circular base.
The height you enter is the vertical height, measured straight down from the tip to the center of the base. It is not the slant height. Mixing the two is the most frequent mistake, and the tool computes the slant height for you, so you should never need to enter it.
The formula πr(r + slant height) combines two parts:
Completing the page's example (radius 3, height 4, slant height 5):
Lateral area = π × 3 × 5 = 15π ≈ 47.12
Base area = π × 3² = 9π ≈ 28.27
Total surface area ≈ 75.40
Take a hypothetical cone with a radius of 5 and a height of 12.
Slant height = √(25 + 144) = √169 = 13
Volume = (1/3) × π × 25 × 12 = 100π ≈ 314.16
Surface area = π × 5 × (5 + 13) = 90π ≈ 282.74
Radius 5 and height 12 form a right triangle with hypotenuse 13, a classic whole-number triple, which makes it easy to verify the slant height by hand.
A cone has exactly one third the volume of a cylinder with the same base radius and height. For radius 3 and height 4, the matching cylinder has volume π × 9 × 4 ≈ 113.10, and one third of that is about 37.70, the same as the cone result. This is a good sanity check: if your cone volume is larger than the volume of the enclosing cylinder, something was entered incorrectly.
It's the distance along the cone's curved surface from the base edge to the apex -- needed to compute the lateral surface area, since the cone's side isn't vertical.
No. Height is the vertical distance from the tip to the center of the base. Slant height is the longer distance along the side, and the calculator works it out for you.
A cone holds exactly one third of the volume of a cylinder with the same radius and height.
Yes. It includes the curved side plus the flat circular base. For an open cone such as a funnel, use only the lateral part, π × r × slant height.
That shape is a frustum. Compute the volume of the full cone, then subtract the volume of the small cone that was removed from the top.
Reference: NIST Weights and Measures