What is the triangle inequality and why does it matter here?

The triangle inequality states each side must be less than the sum of the other two. If violated, no triangle exists and the expression under the square root becomes negative.

When should I use Heron's formula vs the base-height formula?

Use Heron's formula when you know all three sides but no height. Use the base-height formula when you already know a base and its perpendicular height.

Is Heron's formula accurate for very flat triangles?

Standard floating-point arithmetic can lose precision for nearly degenerate triangles. For extreme cases, a numerically stable variant of the formula is recommended.

Triangle Area Calculator - Heron's Formula (3 Sides)

Find the area of any triangle from its three side lengths using Heron's formula. No angles needed. Instant results with step-by-step explanation.

By AbraCalc Editorial Team · Reviewed by AbraCalc Methodology Review · Last reviewed: 2026-06-25

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How to use this tool

Enter side a, side b and side c in the fields above.
Results update instantly as you type — or click Calculate.
Read your area and the full breakdown beneath it.

Formula

s = (a + b + c) / 2; Area = sqrt(s x (s-a) x (s-b) x (s-c))

How it works

Heron's formula computes triangle area from side lengths alone:

Compute the semi-perimeter: s = (a + b + c) / 2
Area = sqrt(s x (s-a) x (s-b) x (s-c))

Worked example

3-4-5 right triangle

s = (3 + 4 + 5) / 2 = 6
Area = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 m^2

6 m^2

Common mistakes to avoid

Entering side lengths that violate the triangle inequality (e.g. sides 1, 2, 10), producing a negative value under the square root.
Using the perimeter instead of the semi-perimeter s; s = (a+b+c)/2, not a+b+c.
Rounding s to the nearest integer before computing the area, introducing significant error.

Key terms

What is the semi-perimeter?: Half the sum of all three side lengths: s = (a+b+c)/2.
When does Heron's formula give zero area?: When the three sides do not form a valid triangle (triangle inequality violated).

Frequently asked questions

What is the triangle inequality and why does it matter here?: The triangle inequality states each side must be less than the sum of the other two. If violated, no triangle exists and the expression under the square root becomes negative.
When should I use Heron's formula vs the base-height formula?: Use Heron's formula when you know all three sides but no height. Use the base-height formula when you already know a base and its perpendicular height.
Is Heron's formula accurate for very flat triangles?: Standard floating-point arithmetic can lose precision for nearly degenerate triangles. For extreme cases, a numerically stable variant of the formula is recommended.