Area of a Triangle (Heron's Formula) Calculator
This calculator finds the area of any triangle using only its three side lengths, so you never need to measure a height or an angle. It uses Heron's formula, which is useful whenever you can measure the sides of a triangle but the perpendicular height is awkward to find, such as with an irregular block of land, a workshop cut-out, or a scalene triangle in a maths problem. You enter the three side lengths, labelled a, b and c, in any consistent unit, and choose a unit label such as cm, m or ft purely for how the results are displayed. The calculator returns the area in square units, along with the semi-perimeter and the full perimeter. It also tells you the triangle type by side length, scalene, isosceles or equilateral, and a full working panel shows the step-by-step calculation: the semi-perimeter, each of s minus a, b and c, the product under the square root, and the final area. A second panel lists the longest side, confirms the triangle inequality, and classifies the triangle as acute, right-angled or obtuse based on its angles. Before calculating, the tool checks that each side is shorter than the sum of the other two; if your three lengths cannot physically form a triangle, it will flag this rather than return a meaningless result.
Triangle Side Lengths
Formula Reference
Heron's formula requires only the three side lengths:
where s is the semi-perimeter. The result is in square units of your side lengths.
The three sides must satisfy the triangle inequality: each side must be less than the sum of the other two.
Step-by-Step Working
Triangle Properties
How Heron's Formula Works
Heron's formula (named after Hero of Alexandria, c. 60 AD) gives the area of any triangle directly from its three side lengths. The first step is to find the semi-perimeter, which is half the triangle's total perimeter:
The area is then:
If all three sides are equal (an equilateral triangle), the formula simplifies to Area = (√3 / 4) × a², but Heron's formula handles every triangle type equally well.
Worked Example
Using the default values: a = 5, b = 6, c = 7.
| Step | Calculation | Result |
|---|---|---|
| Semi-perimeter s | (5 + 6 + 7) / 2 | 9 |
| s − a | 9 − 5 | 4 |
| s − b | 9 − 6 | 3 |
| s − c | 9 − 7 | 2 |
| Product | 9 × 4 × 3 × 2 | 216 |
| Area | √216 | 14.6969 square units |
The perimeter is 5 + 6 + 7 = 18 units. Because all three sides are different lengths, the triangle is scalene.
Triangle Types by Side Length
| Type | Condition | Example |
|---|---|---|
| Equilateral | a = b = c | 3, 3, 3 |
| Isosceles | Exactly two sides equal | 5, 5, 8 |
| Scalene | All three sides different | 5, 6, 7 |
Triangle Inequality
Three lengths can only form a triangle if each side is strictly less than the sum of the other two. For sides a, b, c:
- a + b > c
- a + c > b
- b + c > a
If any condition fails, the three lengths cannot close into a triangle. In Heron's formula this appears as a negative value under the square root, which has no real solution.
Angle Type from Side Lengths
You can determine whether a triangle is acute, right-angled, or obtuse by comparing the square of the longest side to the sum of squares of the other two. Let c be the longest side:
- If c² < a² + b²: acute triangle (all angles less than 90 degrees)
- If c² = a² + b²: right-angled triangle (Pythagorean theorem)
- If c² > a² + b²: obtuse triangle (one angle greater than 90 degrees)
Related Calculators
- Maths and Stats Calculators: full index of geometry and maths tools.
- Area of a Triangle Calculator: area from base and height, or with an angle.
- Triangle Calculator: solve any triangle from sides and angles.
- Pythagorean Theorem Calculator: find a missing side in a right triangle.
- Area Calculator: area formulas for circles, rectangles, and more.
Sources and method: Heron of Alexandria, Metrica (c. 60 AD). The formula is a standard result of Euclidean plane geometry and appears in all major mathematics references including the NZQA Level 2 Mathematics curriculum (AS91259).
This calculator applies Heron's formula exactly. Results are accurate to the precision of your inputs. The tool checks the triangle inequality before computing and will show an error if the three sides cannot form a valid triangle.