# Cake Pan Conversion Calculator

> Computes the surface area (and volume, if a depth is given) of two baking pans of round, square or rectangular shape and the factor by which a recipe must be scaled so the batter reaches the same depth in the new pan, with a baking-time note.

- Calculator id: `cake-pan-conversion` · Category: Food & Cooking (`food`) · Tool name: `convert_cake_pan_size`
- Canonical page: https://tttkmbb.com/food/cake-pan-conversion · This document: https://tttkmbb.com/food/cake-pan-conversion.md · JSON definition: https://tttkmbb.com/food/cake-pan-conversion.json

## Purpose

Computes the surface area (and volume, if a depth is given) of two baking pans of round, square or rectangular shape and the factor by which a recipe must be scaled so the batter reaches the same depth in the new pan, with a baking-time note.

**Use when:** A recipe is written for one pan size and you want to bake it in another, or you need the capacity of a pan in millilitres or US cups.

**Do not use when:** You want to scale a recipe by servings rather than pan size (use recipe-scaler), or the pan is a bundt, tube or oddly shaped tin (measure its capacity with water instead).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `from_shape` | enum: round \| square \| rectangular |  | required | Shape of the pan the recipe was written for. |
| `from_size` | number |  | required | Diameter (round), side (square) or length (rectangular) of the original pan, in size_unit. (> 0, max 200) |
| `from_width` | number |  | optional | Width of the original pan; only for rectangular pans. (> 0, max 200) |
| `to_shape` | enum: round \| square \| rectangular |  | required | Shape of the pan you want to use. |
| `to_size` | number |  | required | Diameter, side or length of the new pan, in size_unit. (> 0, max 200) |
| `to_width` | number |  | optional | Width of the new pan; only for rectangular pans. (> 0, max 200) |
| `size_unit` | enum: cm \| in |  | optional, default "cm" | Unit of all pan dimensions. |
| `depth` | number |  | optional | Depth of the pans in size_unit; when given, pan volumes are reported (the same depth is assumed for both pans). (> 0, max 50) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `scale_factor` | number |  | to_area / from_area: multiply every ingredient by this to get the same batter depth in the new pan. |
| `scale_percent` | number | % | Scale factor as a percentage of the original recipe. |
| `from_area_cm2` | number | cm² | Bottom area of the original pan. |
| `to_area_cm2` | number | cm² | Bottom area of the new pan. |
| `from_area_in2` | number | in² | Same in square inches. |
| `to_area_in2` | number | in² | Same in square inches. |
| `from_volume_ml` | number | ml | area × depth (only when depth is given). |
| `to_volume_ml` | number | ml | area × depth (only when depth is given). |
| `from_volume_cups` | number | US cups | Volume in 236.6 ml US cups (only when depth is given). |
| `to_volume_cups` | number | US cups | Volume in US cups (only when depth is given). |
| `unscaled_depth_ratio` | number |  | Relative batter depth in the new pan when the recipe is used unchanged (from_area / to_area). |
| `baking_note` | string |  | How to scale the recipe and what to expect for baking time. |

## Formula

`area: round = π·(d/2)², square = s², rectangular = l × w; scale_factor = to_area / from_area; volume = area × depth (1 in² = 6.4516 cm², 1 in³ = 16.387 ml, 1 US cup = 236.588 ml)`

Scaling by bottom area keeps the batter depth, and therefore the baking time and crumb, the same; pans should be filled no more than about two-thirds. A 9-inch round (63.6 in²) and an 8-inch square (64 in²) are interchangeable.

## Data Sources

- King Arthur Baking – The essential alternative baking pan sizes — https://www.kingarthurbaking.com/blog/2019/04/23/the-essential-alternative-baking-pan-sizes (reference, retrieved 2026-09-24)
- NIST Special Publication 811 – Guide for the Use of the SI (Appendix B: US customary volume units in litres) — https://www.nist.gov/pml/special-publication-811 (standard, retrieved 2026-09-24)

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/cake-pan-conversion?from_shape=…&from_size=…&to_shape=…&to_size=…`
- `POST https://tttkmbb.com/api/v1/calculate/cake-pan-conversion` 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/cake-pan-conversion · OpenAPI operationId `convert_cake_pan_size` 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": "cake-pan-conversion", "inputs": {…}}`

## Example

- 8 in round → 9 in round, 2 in deep: inputs `{"from_shape":"round","from_size":8,"to_shape":"round","to_size":9,"size_unit":"in","depth":2}` → `{"scale_factor":1.266,"scale_percent":126.6,"from_area_in2":50.3,"to_area_in2":63.6,"from_area_cm2":324.3,"to_area_cm2":410.4,"from_volume_ml":1647,"to_volume_ml":2085,"from_volume_cups":6.96,"to_volume_cups":8.81,"unscaled_depth_ratio":0.79}`
- 20 cm round → 20 cm square: inputs `{"from_shape":"round","from_size":20,"to_shape":"square","to_size":20,"size_unit":"cm"}` → `{"scale_factor":1.273,"from_area_cm2":314.2,"to_area_cm2":400,"to_area_in2":62}`

```
GET https://tttkmbb.com/api/v1/calculate/cake-pan-conversion?from_shape=round&from_size=8&to_shape=round&to_size=9&size_unit=in&depth=2
```

## Limitations

You want to scale a recipe by servings rather than pan size (use recipe-scaler), or the pan is a bundt, tube or oddly shaped tin (measure its capacity with water instead). Scaling by bottom area keeps the batter depth, and therefore the baking time and crumb, the same; pans should be filled no more than about two-thirds. A 9-inch round (63.6 in²) and an 8-inch square (64 in²) are interchangeable. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Do I have to scale the recipe?**

Not for differences under about 10 %. For larger pans the unscaled batter is thinner and bakes faster; for smaller pans it is deeper and needs longer, possibly at a slightly lower temperature, so it sets before the top over-browns.

**How do I convert to a bundt or tube pan?**

Measure the pan's capacity by filling it with water (1 US cup = 237 ml). A 10–12 cup bundt takes the batter of a 9 × 13 in pan and, per King Arthur Baking, bakes about 30 % longer because of the depth.

**Why are the volumes larger than the batter a recipe makes?**

Volumes are for the full pan; batter should fill only one-half to two-thirds of it to leave room for rising.

## Related

- [Recipe Scaler](https://tttkmbb.com/food/recipe-scaler.md) — Apply the scale factor to every ingredient.
- [Oven Temperature Converter](https://tttkmbb.com/food/oven-temperature.md) — Convert oven settings between °C, °F, gas mark and fan ovens.
- [Circle Calculator](https://tttkmbb.com/geometry/circle.md) — Area of round pans.