# Triangle Area Calculator

Area from three sides, with angles, inradius and circumradius.

- Page: https://jianzhujisuan.com/en/triangle-area
- Category: Area and volume (https://jianzhujisuan.com/en/c/area-volume)
- Chinese version: https://jianzhujisuan.com/zh/triangle-area.md
- Data checked: 2026-10-25

## Quick result

With the default inputs (Side a 3 m, Side b 4 m, Side c 5 m): **Triangle area 6.0000 m²**

## Inputs

| Parameter | Type | Default | Unit | Note |
|---|---|---|---|---|
| Side a | number | 3 | m |  |
| Side b | number | 4 | m |  |
| Side c | number | 5 | m |  |

## Formula

- **Heron**: A = √(s(s−a)(s−b)(s−c))
- **Semi-perimeter**: s = (a + b + c) ÷ 2
- **Cosine rule**: cos A = (b² + c² − a²) ÷ 2bc
- **Inradius**: r = A ÷ s
- **Circumradius**: R = abc ÷ 4A
- **Height on a**: ha = 2A ÷ a

## Worked example

Sides of 3, 4 and 5 m: semi-perimeter 6.000 m, area 6.0000 m², angles of 36.87°, 53.13° and 90.00°, a right-angled triangle, inradius 1.0000 m, circumradius 2.5000 m and a height of 2.4000 m on side c.

## Questions

### With only three side lengths, how do I set out the angles?

Use the 3-4-5 method or the cosine rule. 3-4-5 is the practical squareness check: measure 3 m along one leg and 4 m along the other, and if the diagonal reads 5 m the corner is square — within 2 mm is plenty for building work. For other angles compute the third side and strike it from two tapes.

### When does Heron's formula lose accuracy?

On slender triangles where one side approaches the sum of the other two, because s−c becomes a difference of two large similar numbers and most significant digits are lost. For such shapes on site, use base × height ÷ 2 or coordinates — both are far more stable.

### What use are inradius and circumradius on site?

The inradius sets the largest round object that fits inside a triangular area — how big a circular tank fits on a triangular plinth, for instance. The circumradius locates the centre through three columns that are not collinear, which is more accurate on a triangular plot than bisecting diagonals.

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