A cone has a circular base and a single vertex. For a right circular cone, the vertex sits directly above the center of the base. With the base radius and perpendicular height, you can determine volume, slant height, lateral surface area, and total surface area.
What the cone measurements mean
The two dimensions that drive the standard cone formulas are the radius r of the circular base and the height h, measured perpendicular to the base. The slant height s runs from the vertex to the edge of the base. In a right cone, r, h, and s form a right triangle.
Core cone formulas
The base is a circle, so its area is πr². The cone occupies one-third of the volume of a cylinder with the same base and perpendicular height.
V = ⅓πr²hs = √(r² + h²)L = πrsA = πrs + πr²How to calculate cone volume
First find the area of the circular base, πr². Multiply it by the perpendicular height and then divide by 3. The result is expressed in cubic units: cm³, m³, in³, and so on.
So a cone with radius 5 units and height 12 units has a volume of about 314.159 cubic units.
Slant height and surface area
For a right circular cone, the slant height is the hypotenuse of the right triangle formed by the radius and height. Once s is known, the curved lateral surface is πrs. If the circular base is included, add πr².
Right, oblique, and truncated cones
A right cone has its apex directly above the center of its circular base. An oblique cone is tilted, but its volume is still one-third of base area times perpendicular height. A truncated cone, or frustum, has its tip removed by a cut parallel to the base.
| Shape | Volume | Key dimensions |
|---|---|---|
| Right cone | ⅓πr²h | r, h |
| Oblique cone | ⅓πr²h | r, perpendicular h |
| Frustum | ⅓πh(R² + Rr + r²) | R, r, h |
For a frustum, R is the larger base radius and r is the smaller top radius. The calculator above also derives the frustum slant height from √((R−r)²+h²).
Why is a cone one-third of a cylinder?
Take a cone and a cylinder with the same circular base and the same perpendicular height. The cone's volume is exactly one-third of the cylinder's volume. In formula form, the cylinder is πr²h while the cone is ⅓πr²h.
Both solids use the same base area B = πr².
The relevant height is perpendicular to the base plane.
Vcone : Vcylinder = 1 : 3.
Three identical cone volumes fill the matching cylinder volume.
Using the calculator accurately
Enter the base radius and perpendicular height for a right or oblique cone. Choose frustum when the cone has two circular radii. Keep all dimensions in compatible units; if r and h are in centimeters, the volume will be in cubic centimeters and surface areas in square centimeters.
FAQ
For a circular cone, V = ⅓πr²h, where r is the base radius and h is the perpendicular height.
Height is perpendicular to the base. Slant height runs along the cone's side from the vertex to the edge of the base.
Yes. The volume is still one-third of base area times perpendicular height, even when the apex is not directly above the base center.
Use V = ⅓πh(R² + Rr + r²), with R as the larger radius, r as the smaller radius, and h as the perpendicular height.
For a right circular cone, A = πrs + πr². The first term is the curved lateral area and the second is the circular base.