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Geometry

Triangle Area (Heron)

A = √(s(s-a)(s-b)(s-c)). Free online Triangle Area (Heron). Calculate triangle area (heron) online — fast, accurate, mobile-friendly, no signup needed.

Generic triangle with sides a, b, c opposite to angles A, B, C.
Area
6

Derivation

  1. ├── 01Givena = 3, b = 4, c = 5
  2. ├── 02Formula√(r × (r-t) × (r-a) × (r-n))
  3. ├── 03Substitute√(r × (r-t) × (r-3) × (r-n))
  4. └── 04Compute Area6
Did you know?

Pythagoras of Samos (c. 570–495 BCE) likely wasn’t the first to find a²+b²=c² — Babylonian tablets from ~1800 BCE used it — but the first rigorous proof is credited to his school.

§01What is

Understanding the Triangle Area (Heron)

The Triangle Area (Heron) computes Area from 3 inputs: side a, side b, side c. A = √(s(s-a)(s-b)(s-c)).

Geometry is what turns raw measurements into useful answers about space — how much paint, how big a yard, how much material a project will need. Every craftsperson, architect, and DIYer reaches for these formulas regularly. The Triangle Area (Heron) sits in that toolkit — it A = √(s(s-a)(s-b)(s-c)). Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

√(r × (r-t) × (r-a) × (r-n))

Where

a
Side a
b
Side b
c
Side c
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Side a = 3, Side b = 4, Side c = 5.

  1. 01Start by noting the input — Side a: 3.
  2. 02Start by noting the input — Side b: 4.
  3. 03Start by noting the input — Side c: 5.
  4. 04Substitute these values into the formula: √(r × (r-t) × (r-a) × (r-n))
  5. 05Compute Area: the calculator returns 6.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Triangle Area (Heron) Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Side a halved

a = 1.5 (from 3)

Keep every other input at its default and halve the side a. See how area responds.

  1. 01New Side a: 1.5
  2. 02Baseline Area: 6
  3. 03New Area: 2.48039
  4. 04Area decreases by 58.7% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Side a doubled

a = 6 (from 3)

Keep every other input at its default and double the side a. See how area responds.

  1. 01New Side a: 6
  2. 02Baseline Area: 6
  3. 03New Area: 9.92157
  4. 04Area increases by 65.4% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Side b halved

b = 2 (from 4)

Keep every other input at its default and halve the side b. See how area responds.

  1. 01New Side b: 2
  2. 02Baseline Area: 6
  3. 03New Area: 0
  4. 04Area decreases by 100% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Side b doubled

b = 8 (from 4)

Keep every other input at its default and double the side b. See how area responds.

  1. 01New Side b: 8
  2. 02Baseline Area: 6
  3. 03New Area: 0
  4. 04Area decreases by 100% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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