📐 Triangle Area Calculator (Heron Formula)
By ToolNimba Math Team · Updated 2026-06-19
This calculator finds the area of a triangle when you know the lengths of all three sides, using Heron's formula. You do not need the height or any angles: just enter sides a, b and c. The tool first checks that the three lengths can actually form a triangle, then returns the area, the perimeter and the semi-perimeter, along with the working so you can follow each step.
What is the Triangle Area (Heron) Calculator?
Heron's formula gives the area of any triangle from its three side lengths alone. First compute the semi-perimeter s = (a + b + c) / 2, which is just half the perimeter. Then the area is the square root of s(s - a)(s - b)(s - c). It is named after Hero of Alexandria, a Greek engineer from the first century, though the result was likely known earlier. The big advantage is that you never need the height or any angle, which is exactly what you want for a survey plot, a triangular panel, or any shape measured only by its edges.
The method only works for lengths that can form a real triangle. The triangle inequality says each side must be shorter than the sum of the other two: a + b must exceed c, a + c must exceed b, and b + c must exceed a. If any of these fails, the sides cannot close into a triangle, and the product s(s - a)(s - b)(s - c) turns out to be zero or negative, so its square root is not a real area. This calculator flags that case rather than returning a meaningless number.
Heron's formula is one of several ways to find a triangle's area, and which you reach for depends on what you know. If you have a base and the perpendicular height, the simpler 0.5 x base x height is faster. If you know two sides and the angle between them, the formula 0.5 x a x b x sin(C) applies. Heron's formula is the tool of choice when all you can measure are the three edges (the SSS, or side-side-side, case), since it works for every triangle, whether it is acute, right-angled or obtuse.
When to use it
- Finding the area of a triangular plot of land when only the three boundary lengths were measured.
- Working out material for a triangular panel, sail, garden bed or gable when you know the edge lengths but not the height.
- Checking homework or exam answers for the SSS (side-side-side) case in geometry.
- Computing the area of an irregular polygon by splitting it into triangles and measuring each set of sides.
How to use the Triangle Area (Heron) Calculator
- Enter the length of the first side in the Side a box.
- Enter the lengths of the other two sides as Side b and Side c (any consistent unit).
- Optionally type a unit label such as cm or m so the result is labelled.
- Press Calculate area to see the area, perimeter and the step-by-step working.
Formula & method
Worked examples
A right triangle with sides 3, 4 and 5.
- s = (3 + 4 + 5) / 2 = 12 / 2 = 6
- s - a = 6 - 3 = 3, s - b = 6 - 4 = 2, s - c = 6 - 5 = 1
- Area = sqrt(6 x 3 x 2 x 1) = sqrt(36)
- Area = 6
Result: Area = 6 square units, perimeter = 12
A scalene triangle with sides 5, 6 and 7.
- s = (5 + 6 + 7) / 2 = 18 / 2 = 9
- s - a = 9 - 5 = 4, s - b = 9 - 6 = 3, s - c = 9 - 7 = 2
- Area = sqrt(9 x 4 x 3 x 2) = sqrt(216)
- Area ≈ 14.6969
Result: Area ≈ 14.6969 square units, perimeter = 18
Worked areas for sample triangles (Heron's formula)
| Sides (a, b, c) | Semi-perimeter s | Perimeter | Area |
|---|---|---|---|
| 3, 4, 5 | 6 | 12 | 6 |
| 5, 6, 7 | 9 | 18 | 14.6969 |
| 6, 6, 6 | 9 | 18 | 15.5885 |
| 7, 8, 9 | 12 | 24 | 26.8328 |
| 8, 15, 17 | 20 | 40 | 60 |
Common mistakes to avoid
- Entering sides that cannot form a triangle. If one side is equal to or longer than the sum of the other two (for example 2, 3 and 10), no triangle exists. The formula then gives a zero or negative value under the root, so the calculator reports an error instead of a number.
- Mixing different units. All three sides must be in the same unit. Mixing centimetres with metres, or inches with feet, gives a nonsense area. Convert every side to one unit before you calculate.
- Confusing the semi-perimeter with the perimeter. Heron's formula uses s, the semi-perimeter, which is half of a + b + c. Plugging the full perimeter into the formula instead of half of it produces a far too large answer.
- Forgetting the area is in square units. Side lengths are linear, but the area is in square units. If your sides are in metres the area is in square metres, not metres, so label the result accordingly.
Glossary
- Heron's formula
- A formula that gives a triangle's area from its three side lengths, with no need for height or angles.
- Semi-perimeter (s)
- Half of the perimeter, s = (a + b + c) / 2. It is the key intermediate value in Heron's formula.
- Perimeter
- The total distance around the triangle, the sum of all three side lengths a + b + c.
- Triangle inequality
- The rule that each side of a triangle must be shorter than the sum of the other two sides.
- SSS (side-side-side)
- The case where all three side lengths are known, which is exactly what Heron's formula needs.
Frequently asked questions
What is Heron's formula?
Heron's formula finds the area of a triangle from its three side lengths. You compute the semi-perimeter s = (a + b + c) / 2, then the area equals the square root of s(s - a)(s - b)(s - c). It needs no height and no angles.
How do I find the area of a triangle with only the three sides?
Use Heron's formula. Add the sides and halve them to get s, then take the square root of s(s - a)(s - b)(s - c). This calculator does it for you and shows each step, including the perimeter.
What is the SSS case?
SSS stands for side-side-side: the situation where you know all three side lengths of a triangle but no angles or height. Heron's formula is the standard way to get the area in the SSS case.
Why does the calculator say my sides cannot form a triangle?
A real triangle must satisfy the triangle inequality: each side must be shorter than the sum of the other two. If, say, one side is 10 while the others are 2 and 3, the sides cannot close into a triangle, so no area exists.
Does Heron's formula work for all triangles?
Yes. It works for acute, right-angled and obtuse triangles, and for equilateral, isosceles and scalene shapes, as long as the three lengths satisfy the triangle inequality.
What units does the area come out in?
The area is in square units of whatever length unit you enter. If the sides are in centimetres the area is in square centimetres. Add a unit label to have the result labelled automatically.