Cone Volume Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The volume of a cone is calculated with V = (1⁄3) × π × r² × h, where r is the base radius and h the height. For a cone with radius 3 cm and height 4 cm, the volume is about 37.70 cm³.
Explanation
A right circular cone is a solid obtained by connecting all the points of a base circle to an apex located at a certain height above its center. Its volume is exactly one third of that of a cylinder with the same radius and height — a relationship proved by integral calculus, but also verifiable experimentally: a cone must be filled exactly three times to fill a cylinder of the same dimensions. This calculator also gives the slant height (the length of the segment connecting the apex to a point on the base circle) and the lateral area (the area of the conical surface itself, without the base): the slant height is calculated by the Pythagorean theorem from the radius and height, since radius, height, and slant height form a right triangle (see our hypotenuse calculator); the lateral area is then π × radius × slant height. The cone appears frequently in practice: a funnel, a conical hat, a pile of sand or gravel, or an ice cream cone.
Example: cone with radius 3 cm and height 4 cm
Inputs
Radius: 3 cm. Height: 4 cm.
Calculation
Slant height = √(3² + 4²) = √25 = 5 cm. Volume = (1⁄3) × π × 3² × 4 = (1⁄3) × π × 36 = 12π ≈ 37.70 cm³. Lateral area = π × 3 × 5 = 15π ≈ 47.12 cm².
Result
This cone has a volume of about 37.70 cm³ and a lateral area of about 47.12 cm².
Frequently asked questions
Why is a cone’s volume one third of a cylinder’s?
It is a result proved by integral calculus (by summing the area of infinitesimal disks from apex to base), but also very concretely verifiable: filling a cone with water or sand and pouring it three times into a cylinder of the same radius and height fills it exactly. This same one-third proportion applies to any pyramid compared with the prism of the same base and height, not only to the cone.
What is the slant height of a cone?
The slant height is the length of the segment connecting the apex of the cone to any point on the base circle — it corresponds to the oblique line visible on the cone's lateral surface, not to be confused with the height, which is the vertical distance between the apex and the center of the base. Radius, height, and slant height always form a right triangle, which lets any one of these three values be calculated from the other two by the Pythagorean theorem.
How do you calculate the total area of the cone, including the base?
Add the lateral area given by this calculator to the area of the base disk (π × r²): total area = lateral area + π × r². This calculator displays only the lateral area, the most useful in practice when the base does not need to be counted (a conical hat or a cone, for example, have no closed bottom).