How cone volume works
Here is the whole idea in one sentence: a cone holds exactly one third as much as the cylinder it fits inside. Not roughly a third. Exactly a third, every time, for every cone. Take the circular base, find its area with πr², imagine a cylinder of that base rising to the cone's height (base area × height), then cut that volume to a third because the cone tapers to a point instead of staying full-width all the way up.
If that one-third feels arbitrary, try the demonstration that has convinced skeptical students for centuries: get an ice cream cone (a real one, the crunchy kind) and a tin can with the same width and height. Fill the cone with water and pour it into the can. Do it again. Do it a third time. The can is now full, right to the brim. Three cones fill one cylinder, so one cone is one third of a cylinder. The math behind it is a genuine theorem (Cavalieri's principle, or integration if you have calculus), but the kitchen version is the one that sticks.
The formula
Here r is the radius of the circular base, h is the vertical height from the center of the base to the tip, and V comes out in cubic units of whatever unit you measured with. If you have the diameter instead, halve it first: r = d / 2. If you have the slant height s (measured up the outside of the cone), recover the height with the Pythagorean theorem: h = √(s² - r²).
The formula also runs in reverse. Solve for the height when you know the volume: h = 3V / (πr²). Solve for the radius: r = √(3V / (πh)). These come up more often than you might expect, like sizing a funnel to hold a known amount or checking how tall a pile of sand will stand for a given footprint.
Two bonus quantities this calculator gives you for free. The slant height is s = √(r² + h²). The curved side of the cone (lateral surface area) is πrs, and adding the circular base gives the total surface area: πrs + πr².
Worked example
A cone has a radius of 3 and a height of 4.
Volume: V = (1/3) × π × 3² × 4 = (1/3) × π × 9 × 4 = 12π ≈ 37.70 cubic units.
Slant height: s = √(3² + 4²) = √(9 + 16) = √(25) = 5 (the classic 3-4-5 right triangle).
Lateral surface area: πrs = π × 3 × 5 = 15π ≈ 47.12. Total surface area: 15π + π × 3² = 24π ≈ 75.40 square units.
Sanity check: a cylinder with the same radius and height holds π × 9 × 4 = 36π ≈ 113.10, and one third of that is exactly our 12π.
The mistake that triples your answer
The most common cone-volume error is forgetting the 1/3 entirely, which quietly hands you the volume of the surrounding cylinder instead: an answer exactly three times too big. It is worth a five-second sanity check every time: does my answer look like a third of base area times height? The second most common error is plugging the slant height in as the height. The slant height is always longer than the true height (it is the hypotenuse), so this inflates the volume. If you measured up the sloped outside of the cone rather than straight up through the middle, you measured the slant height: use the toggle above and let the calculator recover the true height for you, or compute h = √(s² - r²) yourself first.
Working with other shapes? We have the same treatment for the cylinder and the sphere, and if you just need the base, the circle calculator handles area and circumference.