# Parallelogram Calculator

> Computes the area of a parallelogram from base and height and, when the slanted side is given, its perimeter, the acute angle between base and side in degrees, and both diagonals.

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

## Purpose

Computes the area of a parallelogram from base and height and, when the slanted side is given, its perimeter, the acute angle between base and side in degrees, and both diagonals.

**Use when:** You need the area, perimeter, angles or diagonals of a quadrilateral whose opposite sides are parallel (including rhombi).

**Do not use when:** All angles are right angles (use rectangle) or only one pair of sides is parallel (use trapezoid).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `base` | number | units | required | Length of one side chosen as the base. (> 0) |
| `height` | number | units | required | Perpendicular distance between the base and its opposite side. (> 0) |
| `side` | number | units | optional | Length of the slanted side adjacent to the base (optional; must be at least the height). (> 0) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `area` | number | units² | base × height. |
| `perimeter` | number | units | 2 × (base + side), only when side is given. |
| `angle_deg` | number | degrees | Acute angle between base and side = asin(height / side); the obtuse angle is 180° minus this. |
| `diagonal_long` | number | units | √(base² + side² + 2·base·√(side² − height²)). |
| `diagonal_short` | number | units | √(base² + side² − 2·base·√(side² − height²)). |

## Formula

`area = base × height; perimeter = 2 × (base + side); angle_deg = asin(height / side); diagonals = √(base² + side² ± 2·base·√(side² − height²))`

## Data Sources

- Wikipedia – Parallelogram — https://en.wikipedia.org/wiki/Parallelogram (reference, retrieved 2026-09-23)
- Wolfram MathWorld – Parallelogram — https://mathworld.wolfram.com/Parallelogram.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/parallelogram?base=…&height=…`
- `POST https://tttkmbb.com/api/v1/calculate/parallelogram` 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/parallelogram · OpenAPI operationId `calculate_parallelogram_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": "parallelogram", "inputs": {…}}`

## Example

- Base 8, height 5, side 6: inputs `{"base":8,"height":5,"side":6}` → `{"area":40,"perimeter":28,"angle_deg":56.4427,"diagonal_long":12.372,"diagonal_short":6.8508}`
- Base 10, height 3: inputs `{"base":10,"height":3}` → `{"area":30}`

```
GET https://tttkmbb.com/api/v1/calculate/parallelogram?base=8&height=5&side=6
```

## Limitations

All angles are right angles (use rectangle) or only one pair of sides is parallel (use trapezoid). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Why must the side be at least the height?**

The height is the perpendicular distance between the parallel sides, so the slanted side can never be shorter than it; side = height gives a rectangle.

**How do I compute a rhombus?**

Enter side = base. The area is still base × height, or you can use diagonal_long × diagonal_short / 2.

## Related

- [Rectangle Calculator](https://tttkmbb.com/geometry/rectangle.md) — Special case with right angles.
- [Trapezoid Calculator](https://tttkmbb.com/geometry/trapezoid.md) — Only one pair of parallel sides.
- [Triangle Area Calculator](https://tttkmbb.com/geometry/triangle-area.md) — Half a parallelogram cut along a diagonal.