Heron’s Formula Triangle Area Calculator

This calculator works out the area of a triangle using Heron’s formula. You enter the lengths of all three sides, and you get the area in square units. It also shows the semiperimeter so you can follow along with the calculation.

Formula

Heron’s formula gives the area of a triangle when you know the lengths of all three sides, without needing the height or any angle. In words: first find the semiperimeter (half the perimeter), then multiply it by the difference between the semiperimeter and each side, and take the square root of that product.

In symbols:

Area = √[s(s − a)(s − b)(s − c)]

where:

• a, b, c = lengths of the three sides (in any unit, as long as all three use the same unit)
• s = semiperimeter = (a + b + c) / 2

The result is in square units of whatever unit you used for the sides (e.g., meters → square meters, feet → square feet).

Worked Example

Using the default values from the calculator: sides a = 5 m, b = 6 m, c = 7 m.

1. Calculate the semiperimeter: s = (5 + 6 + 7) / 2 = 18 / 2 = 9 m
2. Subtract each side from s: (s − a) = 9 − 5 = 4; (s − b) = 9 − 6 = 3; (s − c) = 9 − 7 = 2
3. Multiply: s × (s − a) × (s − b) × (s − c) = 9 × 4 × 3 × 2 = 216
4. Take the square root: √216 ≈ 14.6969 m²

Rounded to two decimal places, the area is 14.70 m², which matches the calculator’s output.

When to Use Heron’s Formula

Heron’s formula is the go-to method when you know all three side lengths but do not know the height or any angle. This is common in real-world measurements: you might measure the three sides of a triangular plot of land, a triangular roof section, or a triangular fabric piece, but dropping a perpendicular to measure the height is awkward or impossible. It is also useful when the triangle is scalene (all sides different) and you cannot easily apply the standard ½ × base × height formula.

The formula works for any triangle—acute, obtuse, or right—as long as the three sides actually form a triangle. It is especially handy in surveying, construction, and geometry problems where only side lengths are available.

How to Choose the Right Formula

If you have the base and height, use the simple formula: Area = ½ × base × height. If you have two sides and the included angle, use Area = ½ab·sin(C). If you have all three sides, Heron’s formula is your best choice. You do not need to worry about which side is the base—Heron’s formula treats all sides symmetrically, so the order you enter them does not matter.

One important check: the three sides must satisfy the triangle inequality. The sum of any two sides must be greater than the third side. If not, no triangle exists, and the calculator will show an error or a nonsensical result (like a negative number under the square root).

Common Mistakes and Misconceptions

• Using different units: All three sides must be in the same unit. If one side is in meters and another in centimeters, convert first.
• Forgetting the semiperimeter: A common error is using the full perimeter instead of half. The formula specifically requires s = (a+b+c)/2.
• Mixing up the order of subtraction: The formula subtracts each side from s, not s from each side. Since s is always larger than any side (for a valid triangle), the differences are positive.
• Rounding too early: If you round s or the intermediate products before taking the square root, you may lose precision. Keep full accuracy until the final step.
• Thinking it only works for right triangles: Heron’s formula works for all triangles, not just right ones.

Limitations and Alternatives

Heron’s formula is elegant but not always the most convenient. If you already know the height, the standard base-height formula is simpler. If you know two sides and the included angle, the sine formula is faster. For very large or very small triangles, the formula can be numerically sensitive (due to subtracting nearly equal numbers), but for typical measurements it is fine.

Another alternative is to use coordinates: if you know the vertices of the triangle on a plane, you can use the shoelace formula. But when you only have side lengths, Heron’s formula is the standard and most direct method.

Remember that the area is always positive, and the units are always squared. If you measure sides in inches, the area is in square inches; if in kilometers, square kilometers.

Frequently Asked Questions

Does Heron’s formula work for right triangles?

Yes. Heron’s formula works for any triangle, including right triangles. For a right triangle, it will give the same result as ½ × base × height, but you do not need to identify the height.

What if the sides do not form a triangle?

If the sum of any two sides is less than or equal to the third side, the triangle inequality is violated and no triangle exists. In that case, Heron’s formula will produce a negative number under the square root, which is undefined. The calculator will show an error, and you should recheck your measurements.

Can I use Heron’s formula with sides in different units?

No. All three sides must be in the same unit before applying the formula. If you have mixed units, convert them all to one unit first (e.g., all to meters or all to feet).

Why is it called Heron’s formula?

It is named after Heron of Alexandria, a Greek engineer and mathematician who lived in the 1st century AD. He described the formula in his work Metrica, although some historians believe it may have been known earlier.

What is the semiperimeter?

The semiperimeter is simply half of the triangle’s perimeter. It is denoted by the letter s and is used in Heron’s formula to simplify the expression. For a triangle with sides a, b, c, s = (a+b+c)/2.

How accurate is the calculator?

The calculator uses the exact formula and rounds only the final displayed answer to the number of decimal places you choose. Intermediate calculations are done with full precision, so the result is accurate to the displayed rounding.

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in a | Side a | m | 5 | 0
in b | Side b | m | 6 | 0
in c | Side c | m | 7 | 0
out area | Area | m² | sqrt(((a+b+c)/2)*(((a+b+c)/2)-a)*(((a+b+c)/2)-b)*(((a+b+c)/2)-c)) | 2
out s | Semiperimeter | m | (a+b+c)/2 | 2