Math
Heron's Formula Calculator
Heron's formula gives a triangle's area from its three sides alone — no height or angle needed.
Enter a, b, and c and the calculator shows the semiperimeter, each (s − side) factor, the product under the root, and the area in exact and decimal form, then derives the three heights and angles from it.
All three sides in the same unit. The area comes out in that unit squared.
Try an example
Result
Area
—
- Exact area
- —
- Semiperimeter s
- —
- Heights to a, b, c
- —
- Angles A, B, C
- —
- Perimeter
- —
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Study path
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What Heron's formula solves
The usual area formula ½·base·height needs a height, which you rarely have when only the three sides are given. Heron's formula (from Hero of Alexandria, about 60 AD) uses the sides directly through the semiperimeter s, half the perimeter.
Worked example: sides 7, 8, 9
Sides 7, 8, 9
s = (7 + 8 + 9)/2 = 12. Then s − a = 5, s − b = 4, s − c = 3. Product: 12 · 5 · 4 · 3 = 720. Area = √720 = 12√5 ≈ 26.83 square units.
| Sides | s | Product | Area |
|---|---|---|---|
| 3, 4, 5 | 6 | 6·3·2·1 = 36 | 6 |
| 5, 5, 6 | 8 | 8·3·3·2 = 144 | 12 |
| 7, 8, 9 | 12 | 12·5·4·3 = 720 | 12√5 ≈ 26.83 |
| 10, 10, 10 | 15 | 15·5·5·5 = 1875 | 25√3 ≈ 43.30 |
How to use it
- Enter the three side lengths in the same unit.
- Read the area (exact A root expression written with the √ symbol, such as a square root. when the product is nice, decimal always).
- Follow the steps: semiperimeter, the three subtractions, the product, the square root.
- Use the heights if the next question asks for ½·base·height with a particular base.
How to read the answer
The area is in square units of whatever unit the sides use. The The answer kept as a fraction or radical instead of a rounded decimal. keeps the radical (12√5) when the product under the root is not a perfect square; the decimal is rounded to four places. The heights are the perpendicular distances to each side and always satisfy ½ · side · height = Area.
Common mistakes and edge cases
- Using the full perimeter instead of half of it. s is the semiperimeter.
- Subtracting in the wrong order: it is s − a, not a − s.
- Forgetting to multiply by s itself. There are four factors under the root, not three.
- Sides that fail the triangle inequality (like 2, 3, 6) make the product zero or negative; the calculator reports the valid range for the third side instead.
Worked examples
Sides 7, 8, 9
Area 12√5
Area
26.8328
Sides 3, 4, 5
Right triangle, area 6
Area
6
Equilateral side 10
Area 25√3
Area
43.3013
Decimal sides 2.5, 3.5, 4
Area
4.3301
Impossible 2, 3, 6
Triangle inequality fails
Area
—
Zero side
Area
—
Frequently asked questions
What is the semiperimeter?+
Half the perimeter: s = (a + b + c)/2. Heron's formula uses s and the three differences s − a, s − b, s − c.
Does Heron's formula work for any triangle?+
Yes — right, acute, obtuse, scalene, isosceles, or equilateral — as long as the three sides satisfy the triangle inequality.
Can I get the height from Heron's formula?+
Yes. Once you have the area, the height to side b is h = 2·Area / b. The calculator lists all three heights.
Why is my product negative?+
One of the s − side factors is negative, which means the longest side is longer than the other two combined. Those lengths cannot form a triangle.
How is this different from the triangle area calculator?+
That page uses base and height. This one uses three sides and shows the Heron steps, the exact radical, and the derived heights.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026