Volume and surface of a cone, instantly.
Enter the radius and height of a right circular cone. Read the volume, then the slant height, base area, lateral surface and total surface below, each with the exact formula. The result updates live as you type.
The cone
The radius is the distance from the center of the base to its edge. The height is measured straight up the middle to the tip, not along the slanted side.
What this means
How cone geometry works
The Cone Formulas
A right circular cone is defined by two numbers: the radius r of its circular base and the vertical height h from the base to the tip. From these you get everything else. The volume is V = (1/3) x pi x r^2 x h, exactly one third of a cylinder with the same base and height.
The slant height is the straight distance along the side from the base edge to the tip, l = sqrt(r^2 + h^2). It feeds the surface area: the base area is pi x r^2, the lateral (side) area is pi x r x l, and the total surface area adds the two together, pi x r^2 + pi x r x l.
Slant Height Is Not the Vertical Height
People often mix up the height and the slant height. The height h runs straight up the center from the middle of the base to the tip. The slant height l runs along the outside surface and is always longer, because it is the hypotenuse of a right triangle whose legs are r and h.
For a cone with r = 4 and h = 9, the slant height is sqrt(16 + 81) = sqrt(97), about 9.85. Use h for volume and use l for the surface of the side. Plugging one in for the other is the most common mistake in cone problems.
Where Cones Show Up
Cones are everywhere once you look. An ice cream cone, a traffic cone, a funnel, a party hat, and a paper drinking cup are all right circular cones. A pile of sand or gravel dumped on flat ground naturally forms a cone, so this math estimates stockpile volume for landscaping and construction.
Because the volume is one third of the matching cylinder, a cone holds far less than it looks. That is why a scoop of ice cream sits on top of the cone rather than filling a tall column: the cone itself is a surprisingly small volume.
Common questions
What is the volume of a cone?
The volume is V = (1/3) x pi x r^2 x h, where r is the base radius and h is the vertical height. It equals exactly one third of a cylinder with the same base and height.
How do I find the slant height?
The slant height is l = sqrt(r^2 + h^2). It is the distance along the outside surface from the base edge up to the tip, and it is always longer than the vertical height.
What is the difference between lateral and total surface area?
The lateral surface area, pi x r x l, is only the curved side of the cone. The total surface area adds the flat circular base, pi x r^2, giving pi x r^2 + pi x r x l.
Why is a cone one third of a cylinder?
For the same radius and height, three identical cones exactly fill one cylinder. This is a classic result from calculus and can be shown by pouring water between the shapes, so the cone volume carries the factor of one third.
All geometry runs in your browser. Nothing is uploaded. Results use the exact value of pi and assume a right circular cone, where the tip sits directly above the center of the base.