Find the volume, slant height, and surface areas of a cone. Enter the radius and height.
Usage Tip
Three cones of the same base and height exactly fill one cylinder — that is where the one-third comes from.
slant = √(radius² + height²)
The slant height is the distance from the base edge up to the tip.
Total surface area is the base plus the curved (lateral) area.
Volume is one-third of the matching cylinder.
A cone is described by its base radius and its height. The slant height is the length along the surface from the tip to the edge of the base, found with the Pythagorean theorem as the square root of the radius squared plus the height squared. The volume is one third of the base area times the height, which is exactly one third of the volume of a cylinder with the same base and height.
The lateral surface area is pi times the radius times the slant height, and the total surface area adds the circular base. The slant height, not the vertical height, is what you need for the area of the curved surface. Cones show up in funnels, hoppers, traffic cones, and piles of loose material, where the volume formula gives a quick estimate of quantity.