Triangle Area Calculator
Area from three sides, with angles, inradius and circumradius.
Quick answer
With the default inputs: Side a 3 · Side b 4 · Side c 5Triangle area 6.0000 m²
Edit the inputs below to recalculate.Inputs
Result
Triangle area
6.0000m²
| Perimeter | 12.000m |
| Semi-perimeter | 6.000m |
| Angle A (opposite a) | 36.87° |
| Angle B (opposite b) | 53.13° |
| Angle C (opposite c) | 90.00° |
| Type | Right-angled |
| Inradius | 1.0000m |
| Circumradius | 2.5000m |
| Height on a | 4.0000m |
| Height on b | 3.0000m |
| Height on c | 2.4000m |
Heron needs the triangle inequality to hold — any two sides must exceed the third, otherwise the area is zero. On site, base × height ÷ 2 is more common because the height can be taped; the three-side method suits irregular plots where the height is inaccessible.
Reverse: known result, work out one input
Enter a target result; the other inputs stay fixed and this one is worked out.
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 |
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.
How to work it out
- First check that the three sides form a triangle: any two must sum to more than the third. Sides 3, 4 and 8 do not, so the tool flags an error or returns nothing meaningful.
- Compute the semi-perimeter s = (a + b + c) ÷ 2.
- Apply Heron's formula A = √(s(s−a)(s−b)(s−c)). Given three sides, this is the only route that needs no height.
- Find the angles by the cosine rule: cos A = (b² + c² − a²) ÷ (2bc), then take the inverse cosine. The three angles must sum to 180°, which is the built-in check.
- Inradius r = A ÷ s, circumradius R = abc ÷ (4A). Both are useful for circular components and irregular columns.
- Height on side a is ha = 2A ÷ a, needed for triangular roof frames, trusses and diagonal bracing lengths.
Where it goes wrong
- Applying Heron's formula to three sides without the validity check. A degenerate triangle makes the radicand zero — on site that means three points are collinear and cannot be treated as a triangle.
- Using the wrong included angle in A = 0.5 ab sin C. The angle must be the one between sides a and b, not the angle opposite the third side.
- Not checking that the angles sum to 180°. A mismatch means an input error or the wrong formula, and engineering calculations must verify it.
- Mixing area units. Sides in millimetres give area in mm²; divide by 1e6 for m². This is where precision members go wrong.
Material and cost
Triangles appear in roof frames, trusses, bracing, irregular plots and triangular canopies. Formwork for irregular concrete members is measured on developed area and costs 40-80% more than regular work, with higher labour too.
On site
- For triangular roof areas such as gables and valleys, base × height ÷ 2 is fastest — Heron is unnecessary.
- The 3-4-5 triangle is the classic way to set a right angle on site: measure 3 m, 4 m and 5 m to close the triangle, faster than setting up a theodolite.
- For irregular plots, splitting into triangles is more intuitive than coordinate methods, but keep each triangle as close to equilateral as possible to minimise error.
- Compute truss member lengths trigonometrically rather than scaling off drawings — scale error compounds over long members.
Basis
- GB 50854-2013 Standard method of measurement for building and decorative works
- GB 50017-2017 Standard for design of steel structures, effective lengths
- GB/T 50353-2013 Standard measurement for construction area of building
FAQ
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.
Cite this page
Use this format in reports, method statements or coursework. The link is permanent.
Building Calculator. Triangle Area Calculator. 2026-10-25. https://jianzhujisuan.com/en/triangle-area
Results are estimating aids only and are not a design basis. Construction and final accounts must follow the drawings, the applicable standards and an approved method statement. Anything affecting structural safety, shoring or falsework must be checked by a suitably qualified engineer.