Cone Volume Calculator
Calculate the volume of a cone from its base radius and height in seconds. Perfect for engineers, students, and anyone estimating material quantities for conical shapes like funnels, hoppers, or ice cream cones.
About this calculator
A cone is a three-dimensional solid with a circular base that tapers smoothly to a single apex. Its volume is exactly one-third the volume of a cylinder with the same base radius and height: V = (1/3) × π × r² × h, where r is the base radius and h is the perpendicular height from base to tip. The factor of 1/3 arises from integral calculus: stacking infinitely thin circular discs of decreasing radius from base to apex yields one-third of the bounding cylinder. π (≈ 3.14159) appears because the base is a circle with area πr². This formula applies to right circular cones; for oblique cones the same formula holds as long as h is the true perpendicular height.
How to use
Say you need to fill a conical party hat with confetti. The hat has a base radius of 5 cm and a height of 12 cm. Using the formula: V = (1/3) × π × 5² × 12 = (1/3) × 3.14159 × 25 × 12 = (1/3) × 942.48 ≈ 314.16 cm³. Enter 5 in the Base Radius field and 12 in the Height field. The calculator returns approximately 314.16 cubic units of volume — telling you exactly how much confetti the hat can hold.
Frequently asked questions
How does the volume of a cone compare to the volume of a cylinder?
A cone always holds exactly one-third the volume of a cylinder that shares the same base radius and height. If you filled three identical cones with water and poured them into the matching cylinder, they would fill it perfectly. This 1/3 ratio is a direct consequence of the formula V_cone = (1/3)πr²h versus V_cylinder = πr²h. It is a useful mental shortcut: whenever you know the cylinder volume, divide by three to get the cone volume.
What is the difference between slant height and vertical height in a cone?
The vertical height (h) is the straight-line distance from the apex straight down to the centre of the base — it is perpendicular to the base. The slant height (l) is the distance along the surface from the apex to the edge of the base circle, always longer than h. The two are related by l = √(r² + h²) via the Pythagorean Theorem. For the volume formula you must use the vertical height h, not the slant height. Slant height is used instead when calculating the lateral surface area of the cone.
Why is pi used in the cone volume formula?
Pi (π) enters the formula because the cone's base is a circle, whose area is πr². Since the cone is built by stacking shrinking circles from base to apex, π is an unavoidable part of the calculation. The value π ≈ 3.14159 is the ratio of a circle's circumference to its diameter, and it appears in every formula involving circular or spherical geometry. Without π it would be impossible to convert a radius measurement into the actual area of the circular base, and thus into a meaningful volume.