# Cone Calculator

> Computes the volume, slant height, base area, lateral (curved) surface area, total surface area and half-apex angle in degrees of a right circular cone from its base radius and height.

- Calculator id: `cone` · Category: Geometry (`geometry`) · Tool name: `calculate_cone_properties`
- Canonical page: https://tttkmbb.com/geometry/cone · This document: https://tttkmbb.com/geometry/cone.md · JSON definition: https://tttkmbb.com/geometry/cone.json

## Purpose

Computes the volume, slant height, base area, lateral (curved) surface area, total surface area and half-apex angle in degrees of a right circular cone from its base radius and height.

**Use when:** You need the capacity or surface of a funnel, cone-shaped pile, roof or any right circular cone.

**Do not use when:** The cone is truncated (frustum), oblique, or has a square base (use pyramid).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `radius` | number | units | required | Radius of the circular base. (> 0) |
| `height` | number | units | required | Perpendicular distance from the base to the apex. (> 0) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `volume` | number | units³ | 1/3 × π × r² × h. |
| `slant_height` | number | units | √(r² + h²), distance from the apex to the edge of the base. |
| `base_area` | number | units² | π × r². |
| `lateral_area` | number | units² | π × r × slant_height, the curved surface only. |
| `total_surface_area` | number | units² | π × r × (r + slant_height). |
| `half_apex_angle_deg` | number | degrees | Angle between the axis and the slant side = atan(r / h). |

## Formula

`volume = π · radius² · height / 3; slant_height = √(radius² + height²); base_area = π · radius²; lateral_area = π · radius · slant_height; total_surface_area = π · radius · (radius + slant_height); half_apex_angle_deg = atan(radius / height)`

## Data Sources

- Wikipedia – Cone — https://en.wikipedia.org/wiki/Cone (reference, retrieved 2026-09-23)
- Wolfram MathWorld – Cone — https://mathworld.wolfram.com/Cone.html (reference, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/cone?radius=…&height=…`
- `POST https://tttkmbb.com/api/v1/calculate/cone` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/cone · OpenAPI operationId `calculate_cone_properties` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "cone", "inputs": {…}}`

## Example

- Radius 3, height 4: inputs `{"radius":3,"height":4}` → `{"volume":37.6991,"slant_height":5,"base_area":28.2743,"lateral_area":47.1239,"total_surface_area":75.3982,"half_apex_angle_deg":36.8699}`
- Radius 5, height 12: inputs `{"radius":5,"height":12}` → `{"volume":314.1593,"slant_height":13,"lateral_area":204.2035,"total_surface_area":282.7433,"half_apex_angle_deg":22.6199}`

```
GET https://tttkmbb.com/api/v1/calculate/cone?radius=3&height=4
```

## Limitations

The cone is truncated (frustum), oblique, or has a square base (use pyramid). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Height or slant height?**

Enter the perpendicular height (base to apex along the axis). If you only know the slant height l, the height is √(l² − r²).

**Why is the volume a third of a cylinder's?**

A cone with the same base and height has exactly one third of the cylinder's volume; this follows from integrating the shrinking cross-sections.

## Related

- [Cylinder Calculator](https://tttkmbb.com/geometry/cylinder.md) — Same base without the taper (three times the volume).
- [Square Pyramid Calculator](https://tttkmbb.com/geometry/pyramid.md) — The square-based counterpart.
- [Right Triangle Calculator](https://tttkmbb.com/geometry/right-triangle.md) — The cross-section through the axis is a right triangle with legs r and h.