# Cone Volume Calculator

- **URL:** https://codeasystem.com/calculators/geometry/cone-volume-calculator/
- **Category:** geometry
- **Description:** Get cone volume from radius and height, plus slant height and lateral surface area for material estimates.
- **Primary output:** Volume: 37.7

## Inputs
- Radius (r) (name: `radius`, type: number, example: 3)
- Height (h) (name: `height`, type: number, example: 4)

## Outputs
- Volume: 37.7
- Slant height (l): 5
- Lateral surface area: 47.12
- Total surface area: 75.4

## Formula / methodology
```
V = πr²h⁄3   l = √(r² + h²)   Lateral area = πrl
```

A cone holds exactly one third of the cylinder sharing its base and height; demonstrable by filling three cone-cupfuls into one cylinder. The slant height l comes from the Pythagorean theorem applied to the right triangle formed by radius, height and slope.

Lateral (side) surface area unrolls into a circular sector of radius l and arc length 2πr, giving πrl. Add the base circle πr² for the total enclosing area, shown as an extra row.

## Assumptions & limitations
- Right circular cone with apex directly above the center.
- Perpendicular height, not slant length, goes in the height field.
- Positive lengths in one consistent unit.

## How to use
1. **Enter radius and vertical height**; Height must be perpendicular, not measured along the slope.
2. **Read volume and slant**; Slant height feeds construction: pattern layout, fabric panels, roof framing.
3. **Estimate materials**; Lateral area approximates the outer skin; total surface adds the base disc.

## Example
A cone of radius 3 and height 4 has slant height exactly 5, volume ≈ 37.70 cubic units, and lateral area ≈ 47.12 square units.

Result for these inputs:

```
Volume: 37.7
```

## About this calculator
### The one-third rule

Cones taper linearly from full base to a point, averaging one third of the constant cross-section a cylinder maintains. Ice cream scoopers, sand piles and funnel design all live with this economy; sharp tips cost little material but hold little too.

### Why slant height matters practically

Nobody builds along the invisible vertical axis: fabricators cut patterns along the surface. The 3-4-5 triple hiding in our example (legs r = 3, h = 4 → l = 5) shows how often slant heights come out clean.

## FAQs
### I measured the slant, not the height; now what?

Square the slant, subtract the radius squared, take the root: that converts slant back to vertical height before entering it.

### Does lateral area include the base?

No; it is just the sloped side. The total surface row adds the πr² base disc.

### What about truncated cones (buckets)?

Subtract a smaller cone from a larger one: V = πh(R² + Rr + r²)/3 handles the frustum case.

### Why is my pile’s volume off in practice?

Loose material reposes at its angle of repose and piles are rarely perfect cones; treat results as upper-bound estimates.

## Related calculators
- [Cylinder Volume Calculator](https://codeasystem.com/calculators/geometry/cylinder-volume-calculator/)
- [Sphere Volume & Surface Area Calculator](https://codeasystem.com/calculators/geometry/sphere-volume-surface-calculator/)
- [Pythagorean Theorem Calculator](https://codeasystem.com/calculators/geometry/pythagorean-theorem-calculator/)

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Last updated: 2026-08-23 · Version: 1.0.0 · [HTML version](https://codeasystem.com/calculators/geometry/cone-volume-calculator/)