Cone Volume Calculator ($\mathbf{V = \frac{1}{3}\pi r^2 h}$)
Calculate the volume ($V$), base area ($B$), lateral area ($L$), and slant height ($l$) of a right circular cone. Input the radius ($r$) and the perpendicular height ($h$) below.
Enter Dimensions
Key Geometric Results
Slant Height ($l$)
Base Area ($B$)
Lateral Area ($L$)
Total Surface Area ($A$)
Cone Volume ($V$)
Step-by-Step Solution
Formulas Used in Cone Geometry
All calculations for a right circular cone derive from three fundamental geometric formulas:
1. Cone Volume ($V$)
The volume of a cone is precisely one-third the volume of a cylinder with the same base and height.
Where $r$ is the radius of the base and $h$ is the perpendicular height.
2. Surface Area Formulas
The total surface area ($A$) is the sum of the circular base area ($B$) and the lateral (side) area ($L$).
- **Base Area ($B$):** Area of the circular base. $$\mathbf{B} = \pi r^2$$
- **Lateral Area ($L$):** Area of the slanted side. $$\mathbf{L} = \pi r l$$
- **Total Surface Area ($A$):** $$\mathbf{A} = \pi r^2 + \pi r l$$
3. Slant Height ($l$)
The slant height is necessary for surface area calculations. In a right cone, the radius ($r$), perpendicular height ($h$), and slant height ($l$) form a right triangle. Therefore, we use the Pythagorean theorem:
Right Cone vs. Oblique Cone
This calculator is designed for a **right cone**, where the apex (tip) is vertically aligned over the center of the base, making the height ($h$) perpendicular to the base radius ($r$).
An **oblique cone** has its apex off-center, resulting in a slant height that is not uniform around the circumference. While the volume formula $V = \frac{1}{3} \pi r^2 h$ remains valid if the true perpendicular height ($h$) is known, calculating the surface area of an oblique cone is much more complex.