1. Standard Cone Volume (Radius & Height)
$$V = \frac{1}{3}\pi r^2 h$$
The fundamental formula for the volume of a right circular cone given base radius $r$ and vertical altitude $h$.
Mathematical Reference • Complete Formula Guide
Master Reference of All Cone Formulas
A definitive mathematical index of formulas for right circular cones, conical frustums, oblique cones, hollow shells, and surface area. Each formula includes variable definitions, mathematical explanations, and direct links to our dedicated calculators.
2. Cone Volume from Diameter
$$V = \frac{1}{12}\pi d^2 h \approx 0.2618 \cdot d^2 h$$
Direct volume formula using base diameter $d = 2r$. Eliminates rounding errors from preliminary division.
3. Cone Volume from Slant Height
$$V = \frac{1}{3}\pi r^2 \sqrt{l^2 - r^2}$$
Solves volume when slant height $l$ is measured instead of vertical height, substituting $h = \sqrt{l^2 - r^2}$.
4. Cone Volume from Circumference
$$V = \frac{C^2 h}{12\pi}$$
Calculates volume when base perimeter $C = 2\pi r$ is measured with a flexible tape measure ($r = C / 2\pi$).
5. Cone Volume from Base Area
$$V = \frac{1}{3} B h$$
Universal formula for any cone or pyramid when the cross-sectional base area $B$ and perpendicular height $h$ are known.
6. Slant Height Formula (Pythagorean)
$$l = \sqrt{r^2 + h^2}$$
Determines the generatrix length from the apex to the circular edge using right-triangle geometry.
7. Lateral Surface Area (Curved Face)
$$A_{lat} = \pi r l = \pi r \sqrt{r^2 + h^2}$$
The surface area of the cone's sloped walls excluding the circular bottom base.
8. Total Surface Area
$$A_{tot} = \pi r^2 + \pi r l = \pi r(r + l)$$
Combined area of the circular base ($B = \pi r^2$) plus the curved lateral mantle ($A_{lat} = \pi r l$).
9. Truncated Cone (Frustum Volume)
$$V = \frac{1}{3}\pi h (R^2 + r^2 + Rr)$$
Volume of a truncated cone with bottom radius $R$, top radius $r$, and vertical height $h$. Essential for hoppers and lampshades.
10. Oblique Cone Volume
$$V = \frac{1}{3}\pi r^2 h$$
By Cavalieri's principle, an oblique (tilted) cone has the exact same volume as a right cone of identical base and vertical height $h$.
11. Elliptical Cone Volume
$$V = \frac{1}{3}\pi a b h$$
Volume of an oval/elliptical cone where base semi-major axis is $a$ and semi-minor axis is $b$ (Base Area $B = \pi a b$).
12. Hollow Cone Volume (Casing)
$$V = \frac{1}{3}\pi h (R^2 - r^2)$$
The solid material volume of a conical shell with outer radius $R$, inner radius $r$, and height $h$.
13. Double Cone Volume (Hourglass)
$$V = \frac{2}{3}\pi r^2 h$$
Two identical cones joined at their apex or bases ($2 \times \frac{1}{3}\pi r^2 h$).
14. Cone with Vertex Angle ($\theta$)
$$V = \frac{1}{3}\pi r^3 \cot(\theta)$$
Computes volume using base radius $r$ and half-apex angle $\theta$, substituting $h = r \cot(\theta)$.
Quick Reference
Cone Formulas Cheat Sheet
Summary comparison of parameters, variables, and units.
| Geometric Quantity | Standard Formula | Known Parameters | Standard Units | Dedicated Tool |
|---|---|---|---|---|
| Volume ($V$) | $V = \frac{1}{3}\pi r^2 h$ | Radius $r$, Height $h$ | cm³, m³, in³, L | Volume Solver |
| Volume from Diameter | $V = \frac{1}{12}\pi d^2 h$ | Diameter $d$, Height $h$ | cm³, m³, gal | Diameter Solver |
| Slant Height ($l$) | $l = \sqrt{r^2 + h^2}$ | Radius $r$, Height $h$ | cm, m, in, ft | Slant Solver |
| Lateral Surface Area | $A_{lat} = \pi r l$ | Radius $r$, Slant $l$ | cm², m², in² | Area Solver |
| Total Surface Area | $A = \pi r(r + l)$ | Radius $r$, Slant $l$ | cm², m², ft² | Area Solver |
| Frustum Volume | $V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)$ | $R, r, h$ | cm³, m³, L, gal | Frustum Solver |