Triangle Area Calculator (3 Sides)
This calculator finds the area of a triangle when you only know the length of its three sides, with no angle or height measurement needed. It applies Heron's formula, the standard method whenever a right angle or a known base and height are not available, such as when you have measured a plot of land, a fabric panel, or a garden bed with a tape measure alone. Enter the lengths of side a, side b, and side c in any consistent unit, and the calculator works through the two-step formula automatically, first the semi-perimeter, then the area itself. Alongside the area, you get the perimeter, the semi-perimeter, and the triangle type, equilateral, isosceles or scalene, based on your entered sides, plus a full step-by-step breakdown of every intermediate value and the height measured from the longest side. Before it calculates anything, the tool checks the triangle inequality, that the sum of any two sides must exceed the third, and flags an error if your figures cannot form a real triangle. Use it to check a geometry answer, plan a cut of material, or work out how much space a triangular section of land or garden covers. Results are rounded for display, so treat tiny differences in the final decimal place as a rounding effect rather than an error in the method.
Enter the Three Side Lengths
Heron's Formula
s = (a + b + c) / 2Step 2: Area
= sqrt(s × (s−a) × (s−b) × (s−c))Enter your side lengths to see the working.
The result is in square units matching your input (e.g. if sides are in centimetres, area is in cm²).
Step-by-Step Calculation
Triangle Properties
How to Calculate Triangle Area from Three Sides
When you know all three sides of a triangle but not the height or any angles, Heron's formula is the standard method. It was described by Hero of Alexandria around 60 AD and works for any triangle, regardless of whether it is acute, right-angled, or obtuse.
The process has two steps:
- Calculate the semi-perimeter: s = (a + b + c) / 2
- Calculate the area: Area = sqrt(s x (s - a) x (s - b) x (s - c))
Worked Example
For a triangle with sides a = 5, b = 6, and c = 7:
| Step | Calculation | Result |
|---|---|---|
| Semi-perimeter | s = (5 + 6 + 7) / 2 = 18 / 2 | 9 |
| s - a | 9 - 5 | 4 |
| s - b | 9 - 6 | 3 |
| s - c | 9 - 7 | 2 |
| Product | 9 x 4 x 3 x 2 | 216 |
| Area | sqrt(216) | 14.6969 sq units |
These are the default values shown in the calculator above, confirming that the result is 14.70 square units (to 2 decimal places).
Triangle Types by Side Length
| Type | Side Lengths | Example |
|---|---|---|
| Equilateral | All three sides equal | a = b = c = 5 |
| Isosceles | Exactly two sides equal | a = b = 5, c = 7 |
| Scalene | All three sides different | a = 5, b = 6, c = 7 |
The Triangle Inequality
Not every set of three lengths forms a valid triangle. For sides a, b, and c to form a triangle, all three of these conditions must hold:
- a + b > c
- a + c > b
- b + c > a
If any condition is not met, the calculator will display an error. For example, sides 1, 2, and 10 fail because 1 + 2 = 3 which is not greater than 10.
Height from the Longest Side
Once you have the area, you can find the height (altitude) from any base using the formula: height = (2 x Area) / base. The calculator shows the height measured from the longest side. This is useful for checking your answer or for practical tasks like cutting timber or marking out shapes on the ground.
Related Calculators
- Maths and Stats Calculators: all maths tools on this site.
- Area of a Triangle Calculator: base and height method.
- Area of a Triangle (Heron) Calculator: extended Heron's formula tool.
- Triangle Calculator (general): solve any triangle from mixed inputs.
- Pythagorean Theorem Calculator: right-triangle side lengths.
Method: Heron's formula. Hero of Alexandria, Metrica, c. 60 AD. Standard formula reproduced in any geometry textbook (e.g. Coxeter, Introduction to Geometry, Wiley). The triangle inequality is a fundamental theorem in Euclidean geometry.
This calculator uses exact arithmetic (JavaScript's IEEE 754 double precision). Results are rounded for display. For very small or very large triangles, verify with an independent source if precision is critical.