Heron's Formula Calculator

Calculate the area of a triangle from its three side lengths using Heron's formula, with the semi-perimeter and a validity check.

Inputs

Input format and limits

Nonzero magnitudes below 1e-50 are outside this floating-point workspace.

Current result

Active result

Coordinate table for the diagram

3 min read 3 steps 6 terms 3 examples 6 FAQs s = (a + b + c) / 2

Heron's formula lets you find the area of any triangle using nothing but its three side lengths — no angles, no height measurement required.

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Walk-through

How to Use This Calculator

3 steps

Choose a workflow

Select the required known-input mode: Ordinary triangle; Degenerate limit preview; Measurement ranges. Only fields for the selected workflow affect the calculation.

Inspect the current snapshot

Use finite inputs ≤1e50 in magnitude and angles ≤1e12°. Accept pi, sqrt(), fractions and arithmetic; inspect formulas and coordinate tables. Geometry alone does not establish physical safety.

Save, restore and export

Save/Restore uses a local v2 snapshot of inputs, mode, units, precision and selected result. Share replays it; Copy/CSV include all results. Invalid input clears results and disables result exports.

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Reference

Formula & Methodology

3 formulas

Semi-perimeter

s = (a + b + c) / 2

Half the triangle's perimeter. It's an intermediate value with no independent geometric meaning — it exists purely to feed into Heron's formula below.

Heron's formula (area)

Area = √(s(s−a)(s−b)(s−c))

The area of a triangle computed directly from its three side lengths a, b, and c, using the semi-perimeter s. Named for Heron of Alexandria, who described it roughly 2,000 years ago.

Triangle inequality (validity)

a+b > c, a+c > b, b+c > a

All three of these must hold for a, b, and c to form a real, non-degenerate triangle. If any side is greater than or equal to the sum of the other two, no triangle exists and the area formula would require taking the square root of a negative number.

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Glossary

Key Terms Explained

6 terms

Heron's formula A method for computing the area of a triangle from its three side lengths alone: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter. It works for any triangle, right or oblique.

Semi-perimeter Half of a triangle's perimeter: s = (a+b+c)/2. Used as an intermediate step inside Heron's formula.

Triangle inequality The rule that in any valid triangle, each side must be shorter than the sum of the other two sides (a+b>c, a+c>b, b+c>a). If this fails, the three lengths cannot enclose a triangle.

SSS (side-side-side) A way of specifying a triangle using only its three side lengths, with no angles required. Heron's formula is the standard way to find the area of an SSS triangle.

Area The amount of two-dimensional space enclosed by the triangle's three sides, expressed in squared units (e.g. square feet, square meters) matching whatever unit the side lengths were entered in.

Degenerate triangle A set of three side lengths that fails the triangle inequality (or exactly equals it, forming a flat line with zero area) and therefore cannot form a real, enclosed triangle.

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Scenarios

Real-World Examples

3 worked examples

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Classic right triangle

Sides 3, 4, 5

Side a 3 Side b 4 Side c 5

This is the well-known 3-4-5 right triangle. The semi-perimeter is s = (3+4+5)/2 = 6, so the area is √(6×3×2×1) = √36 = 6 — matching the familiar ½×3×4 = 6 you'd get from the right-triangle base/height formula.

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Equilateral triangle

Sides 6, 6, 6

Side a 6 Side b 6 Side c 6

With all three sides equal to 6, the semi-perimeter is s = 9, and the area works out to √(9×3×3×3) = √243 ≈ 15.59. Every equilateral triangle's area can be found this way without ever needing to compute a height.

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Invalid triangle

Sides 2, 2, 5

Side a 2 Side b 2 Side c 5

Here 2+2=4, which is less than the third side (5), so the triangle inequality fails — these three lengths can never meet up to enclose a triangle. The Validity calculation view flags this immediately, and the Area calculation view shows "Invalid" rather than attempting to compute a square root of a negative number.

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Reference

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Deep Dive

Understanding Heron's Formula

4 chapters · 3 min read

Heron's formula lets you find the area of any triangle using nothing but its three side lengths — no angles, no height measurement required. It's named after Heron of Alexandria, who documented it roughly 2,000 years ago, though the result may be even older.

How Heron's formula works

Given three side lengths a, b, and c, the formula first computes the semi-perimeter s = (a+b+c)/2 — literally half the perimeter. The area then follows from Area = √(s(s−a)(s−b)(s−c)). Each of the terms (s−a), (s−b), and (s−c) is positive only when the triangle inequality holds, which is exactly why an invalid set of side lengths produces a negative number under the square root instead of a real area.

Why it's useful

Most area formulas — like ½ × base × height — require you to know or measure a perpendicular height, which isn't always easy to find directly, especially for oblique (non-right) triangles. Heron's formula sidesteps that entirely: if you can measure the three sides (with a tape measure, survey data, or GPS coordinates converted to distances), you can get the area without ever finding an angle or a height.

Inputs and what they mean

The calculator takes three inputs — side a, side b, and side c — all in the same unit (feet, meters, inches, etc.). The order doesn't matter: Heron's formula is symmetric in a, b, and c, so labeling them differently just relabels which side is which without changing the result. The resulting area is always in squared units of whatever unit you entered.

Limits and edge cases

Heron's formula only produces a real, positive area when the three lengths satisfy the triangle inequality (each side shorter than the sum of the other two). If they don't, no triangle exists, and the calculator's Validity calculation view will flag exactly which inequality failed. A borderline case — where one side exactly equals the sum of the other two — describes a degenerate "triangle" that has collapsed into a straight line with zero area.

Use finite values with magnitude at most 1e50. Expressions support pi, sqrt(), fractions and arithmetic. Angles are limited to 1e12 degrees. Units are mathematical; geometry alone does not establish physical safety. Read the linked values, named formulas, diagram coordinate table and any branch or parameter comparison.

Save scenario stores version 2 inputs, workflow, units, precision and selected result on this device. Restore saved scenario replays that snapshot. Share uses the same state; Copy and CSV include the selected answer and all current derived values. Invalid inputs clear the result and disable result exports.

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Questions

Frequently Asked Questions

6 questions

What is the formula for Heron's formula?+

Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2 and a, b, c are the three side lengths.

Do I need all 3 sides to use Heron's formula?+

Yes — Heron's formula is an SSS (side-side-side) method. You need all three side lengths and nothing else.

What makes a triangle invalid?+

A triangle is invalid when the triangle inequality fails: one side is greater than or equal to the sum of the other two (a+b≤c, a+c≤b, or b+c≤a).

How is Heron's formula different from ½ × base × height?+

½bh needs a perpendicular height, which isn't always convenient to measure. Heron's formula needs no height at all — just the three side lengths.

What is the semi-perimeter?+

The semi-perimeter is half the triangle's perimeter: s = (a+b+c)/2. It's an intermediate value used only to compute the area.

What units is the area in?+

Squared units matching whatever unit you entered for the sides — e.g. side lengths in feet give an area in square feet.

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