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.

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.

Frequently Asked Questions (FAQ)

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V = \frac{1}{3} B h = \frac{1}{3} \pi r^2 h

\mathbf{B} = \pi r^2

\mathbf{L} = \pi r l

\mathbf{A} = \pi r^2 + \pi r l

l = \sqrt{r^2 + h^2}

$V = \frac{1}{3} B h = \frac{1}{3} \pi r^2 h$

$l = \sqrt{r^2 + h^2}$