Cone Volume Calculator

Volume of a cone is V = (1/3) × π × r² × h. A cone has exactly one-third the volume of the cylinder that would just enclose it — same base radius, same height. The factor of one-third comes from integral calculus: the cone's cross-section shrinks linearly from the base to a point at the apex, and the integral of a linear function from 0 to its maximum is one-half the maximum, which combined with the geometric setup yields the (1/3) factor.

Surface area splits into two parts. Lateral (curved) surface = π × r × ℓ, where ℓ is the slant height, calculated by Pythagoras: ℓ = √(r² + h²). Base = π × r². Total = π × r × (r + ℓ). The base is excluded for open cones like ice-cream cones; the calculator shows both so you can pick the relevant number for your job.

Engineers estimate the volume of conical stockpiles (sand, gravel, salt) by surveying base radius and peak height. A salt pile 30 feet in radius and 20 feet tall holds (1/3) × π × 900 × 20 ≈ 18,850 cubic feet — about 1,500 tons at 80 lb/ft³. Off-road truck operators rely on these estimates daily; an error of 10% in the peak-height measurement turns into a 10% error in tonnage, which translates directly to a six-figure inventory discrepancy at municipal sand depots.

A standard paper cone cup has a top diameter of 3 inches and a height of 4 inches. Radius = 1.5 in. Volume = (1/3) × π × 1.5² × 4 ≈ 9.42 in³, which is about 5.1 fl oz — slightly more than the nominal '5 oz' printed on the cup. Slant height is √(1.5² + 4²) = √18.25 ≈ 4.27 in, which is the length of the flat sector of paper used to roll the cup at the factory.

Oblique cones. The volume formula V = (1/3) × π × r² × h still holds for tilted cones as long as h is the perpendicular height from the base to the apex, not the slant. Surface area formulas, however, only apply to right circular cones.

Frustums (truncated cones). A cone with the top sliced off has volume V = (1/3) × π × h × (R² + R × r + r²), where R is the larger radius and r is the smaller. Most drinking cups are frustums, not full cones.

Cubic units. Volume is always in cubed units of the input — if you measured radius in feet and height in inches without converting, the answer is meaningless. Pick one unit and stick with it.

Surface area uses slant height, not vertical height. Mixing them up understates the lateral area by anywhere from 5% to 50% depending on the cone's proportions.