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Heron's Formula Calculator

Calculate the area of a triangle from all three side lengths using Heron's formula.

By Christopher FloiedPublished Updated

Results

Area

14.6969 sq units

Semi-perimeter

9 units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Heron's Formula Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Heron's Formula Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Heron's Formula Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Heron's Formula Calculator

Heron's formula is a remarkable result that allows you to calculate the area of any triangle when you know only the three side lengths — no need for the height or any angles. Named after Hero of Alexandria (c. 10-70 AD), the formula states that A = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (half the perimeter). This formula is invaluable in surveying, where side lengths can be measured directly with a tape but heights are difficult to determine. It is also used extensively in computational geometry, computer graphics, and finite element analysis. Heron's formula works for all valid triangles — acute, right, and obtuse — making it one of the most versatile area formulas in geometry. This calculator computes both the area and the semi-perimeter.

The Math Behind It

Heron's formula can be derived from the standard area formula A = ½ab sin(C) combined with the law of cosines. From the law of cosines, cos(C) = (a² + b² − c²)/(2ab), so sin(C) = √(1 − cos²(C)). Substituting and simplifying yields Heron's formula. An alternative derivation uses the identity A = ¼√(4a²b² − (a² + b² − c²)²), which factors into the Heron form. The formula can also be expressed as A = ¼√((a+b+c)(−a+b+c)(a−b+c)(a+b−c)). For the formula to yield a real area, the three sides must satisfy the triangle inequality: a + b > c, a + c > b, and b + c > a. If any factor under the square root is negative, the sides cannot form a triangle. Numerically, Heron's formula can suffer from cancellation errors when one side is much smaller than the others. In such cases, Kahan's formula provides better numerical stability: sort sides so a ≥ b ≥ c, then A = ¼√((a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))).

Formula Reference

Heron's Formula

A = √(s(s−a)(s−b)(s−c))

Variables: s = (a+b+c)/2 (semi-perimeter), a, b, c = side lengths

Worked Examples

Example 1: Land survey

A triangular plot has sides 150 m, 200 m, and 250 m.

Step 1:s = (150 + 200 + 250) / 2 = 300
Step 2:A = √(300 × (300−150) × (300−200) × (300−250))
Step 3:A = √(300 × 150 × 100 × 50)
Step 4:A = √(225,000,000)

A = 15,000 m²

Common Mistakes & Tips

  • !Forgetting to compute the semi-perimeter (using the full perimeter instead).
  • !Not checking the triangle inequality — sides that do not form a valid triangle give imaginary results.
  • !Numerical errors from floating-point cancellation when one side is very small relative to the others.

Related Concepts

Triangle Area

The simpler A = ½bh formula when height is known.

Law of Cosines

Can be used to find angles when all sides are known.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Triangle Area Calculator

Frequently Asked Questions

Can Heron's formula give a negative value under the square root?

Yes, if the sides do not satisfy the triangle inequality (a + b > c for all permutations). This means the sides cannot form a valid triangle.

Is Heron's formula equivalent to A = ½bh?

Yes. Both give the same area. Heron's formula is derived from A = ½bh by expressing h in terms of the three sides using the Pythagorean theorem.

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