✦ Interactive Geometry Suite & Solvers

The Definitive Cone Volume Formula & 3D Visualizer

Calculate the exact volume, slant height, lateral surface area, and base dimensions of any right cone or frustum. Discover step-by-step mathematical proofs, real-time 3D models, and multi-unit conversions.

Primary Mathematical Formula
$$V = \frac{1}{3}\pi r^2 h = \frac{\pi d^2 h}{12}$$
V = Volume r = Radius d = Diameter h = Vertical Height
Launch Calculator Watch Why 1/3 Proof
16 Dedicated Calculators
100% Step-by-Step Proofs
11+ Global Unit Formats
Interactive Preview Live 3D Render
All-in-One Geometry Suite

Cone Volume Calculator with Real-Time 3D Modeling

Choose your known parameters below. Adjust dimensions using input fields or sliders to watch the cone and step-by-step arithmetic update dynamically in real time.

Standard Cone Volume ($r$ & $h$)

Enter the base radius and vertical height to calculate volume and surface area.
r
h
h: 12.00 cm
r: 5.00 cm
l: 13.00 cm
🖱️ Click & Drag to Rotate in 3D
Calculated Cone Volume ($V$)
314.1593 cm³
Exact Form: 100.0000 π cm³ ≈ 314.1593 cm³
Slant Height ($l$)
13.0000 cm
$\sqrt{r^2 + h^2}$
Base Surface Area ($B$)
78.5398 cm²
$\pi r^2$
Lateral Area ($A_{lat}$)
204.2035 cm²
$\pi r l$
Total Surface Area ($A_{tot}$)
282.7433 cm²
$\pi r(r + l)$
Dedicated Geometry Solvers

Explore Other Cone Calculators

Need a specialized solver for slant heights, tank hoppers, liquid dip fill levels, or non-standard frustums? Choose from our 16 dedicated, step-by-step cone calculators below.

Looking for a Complete Formula Reference & Engineering Hub?

Visit our central Cone Calculators Suite Hub featuring formula breakdowns, geometry cheat sheets, and technical FAQs across all 16 cone solvers.

Explore All 16 Cone Calculators Hub
Interactive Geometric Proof

Why is the Volume of a Cone Exactly $\frac{1}{3}$ of a Cylinder?

Experience the famous physical water pouring experiment discovered by Archimedes and verified by calculus integration.

Water Pouring Experiment Simulation

Click the button below to pour water from 3 identical cones into the same-size cylinder ($r$ and $h$ equal).

Full Cone ($V_{cone}$)
Ready to Pour
Target Cylinder ($V_{cylinder} = \pi r^2 h$)
1/3 2/3 Full
Current Fill: 0 / 3 Cones (0%)

Mathematical Derivations

Method 1: Disk Integration (Calculus)

Consider a cone of height $h$ and base radius $r$ oriented along the $z$-axis with the apex at the origin $(0,0,0)$. At any height $z$ from the apex, the cross-sectional slice is a circle whose radius $r(z)$ is given by similar triangles:

$$r(z) = \frac{r}{h} \cdot z$$

The area of each circular disk slice is $A(z) = \pi [r(z)]^2 = \pi \left(\frac{r}{h}z\right)^2 = \frac{\pi r^2}{h^2} z^2$. Integrating these slices from $z = 0$ to $z = h$:

$$V = \int_{0}^{h} A(z) \, dz = \frac{\pi r^2}{h^2} \int_{0}^{h} z^2 \, dz = \frac{\pi r^2}{h^2} \left[ \frac{z^3}{3} \right]_0^h$$ $$V = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3} = \frac{1}{3}\pi r^2 h$$

✦ The factor $\frac{1}{3}$ directly results from the fundamental anti-derivative $\int z^2 dz = \frac{z^3}{3}$!

Method 2: Cavalieri's Principle & Pyramids

A cone is a circular pyramid. Any pyramid with base area $B$ and height $h$ has volume $V = \frac{1}{3} B h$. Because a cube can be dissected into 3 congruent pyramids whose vertices meet at a center, all conical solids inherit the exact $\frac{1}{3}$ ratio!

Formula Reference & Cheat Sheet

All Cone Geometry Formulas

A comprehensive reference for examinations, homework, and engineering calculations.

Primary Volume ($V$)
$$V = \frac{1}{3}\pi r^2 h$$

Computes the total capacity or cubic space enclosed by a right circular cone.

$r$ = Radius of base circle
$h$ = Vertical perpendicular height
Diameter Volume ($d$)
$$V = \frac{\pi d^2 h}{12}$$

Useful when measuring pipe, cup, or hopper diameters directly ($d = 2r \Rightarrow r^2 = d^2/4$).

$d$ = Full base diameter ($2r$)
$h$ = Vertical perpendicular height
Pythagorean Slant ($l$ or $s$)
$$l = \sqrt{r^2 + h^2}$$

Length of the hypotenuse from any point on the base perimeter up to the apex.

$l$ = Slant height / generatrix
$h = \sqrt{l^2 - r^2}$ (rearranged)
Surface Lateral ($A_L$)
$$A_L = \pi r l = \pi r \sqrt{r^2 + h^2}$$

The curved exterior side surface area excluding the flat circular base.

$l$ = Slant height
Equivalent to an unwrapped circular sector
Full Area Total ($A_T$)
$$A_T = \pi r^2 + \pi r l = \pi r(r + l)$$

Base area plus lateral area. Total material required to construct a closed solid cone.

$\pi r^2$ = Base Area
$\pi r l$ = Lateral Area
Truncated Frustum ($V$)
$$V = \frac{1}{3}\pi h (R^2 + r^2 + Rr)$$

A cone with its apex sliced off parallel to the base (like a standard drinking cup or lampshade).

$R$ = Bottom base radius
$r$ = Top base radius
Frequently Asked Questions

Common Questions About Cone Volume

Clear answers to top geometry questions asked by students, educators, and engineers.

The standard formula is $V = \frac{1}{3}\pi r^2 h$, where $r$ is the radius of the circular base, $h$ is the perpendicular vertical height from the base to the apex point, and $\pi \approx 3.14159$. If using diameter ($d$), the formula becomes $V = \frac{\pi d^2 h}{12}$.

The vertical height ($h$) is the perpendicular distance straight down from the apex to the center of the base circle at a 90° angle. The slant height ($l$) is the distance along the outer surface from the apex to the edge of the circular base. They form a right-angled triangle with the radius: $l^2 = r^2 + h^2$. Slant height is used for surface area, while vertical height is used for volume.

First calculate the perpendicular vertical height $h$ using the Pythagorean theorem: $h = \sqrt{l^2 - r^2}$. Once you have $h$, plug $r$ and $h$ into the standard volume equation $V = \frac{1}{3}\pi r^2 h$. You can also select the "Radius & Slant Height" tab in our calculator above to compute this automatically.

A frustum is a cone whose top portion (including the apex) has been cut off by a plane parallel to the base, resulting in two circular flat faces of different radii ($R$ and $r$). Its volume formula is $V = \frac{1}{3}\pi h(R^2 + r^2 + R \cdot r)$, where $h$ is the vertical height between the two circular faces.

Yes! By Cavalieri's Principle, any two solids of equal height with cross-sectional slices of equal areas have identical volumes. Therefore, the volume formula $V = \frac{1}{3}\pi r^2 h$ applies to both right cones and oblique cones, provided that $h$ is the true vertical perpendicular height between the base plane and the apex point.

$1\text{ cm}^3 = 1\text{ mL} = 0.001\text{ Liters}$. To convert $cm^3$ to Liters, divide by 1,000. For US liquid gallons, $1\text{ gallon} \approx 3,785.41\text{ cm}^3$ (or $0.13368\text{ ft}^3$). Our calculator above automatically converts results into all metric and imperial units in real time.