Triangle Area from Three Sides Calculator

Formula

Heron's formula

Worked example

With Side a: 3 units, Side b: 4 units, Side c: 5 units, area is 6 square units.

Calculation method

The rule is Heron's formula.

Heron’s formula first takes the semiperimeter s = (a+b+c)/2, then gives area A = √[s(s−a)(s−b)(s−c)]. The three lengths must satisfy the triangle inequality.

For this triangle area from three sides example, changing side a from 3 units to 4 units while keeping the other inputs fixed changes area from 6 square units to 7.806247 square units.

Reading area

The result is an area in square units. The same three side lengths define the same area regardless of how the triangle is rotated. In the worked values on this page, side a 3 units, side b 4 units, side c 5 units produce area 6 square units.

Practical use

Use it when the three side lengths are known but the perpendicular height is not. The specific task here is to find area from side a, side b, side c.

Limits and common mistakes

The three lengths must satisfy the triangle inequality: each side must be shorter than the sum of the other two. Verify side a in units before interpreting area; a value measured on another basis would answer a different question.

Why the relationship works

Heron’s formula uses all three side lengths. First calculate semiperimeter as half their sum, then combine it with the three differences between semiperimeter and each side. No separate height measurement is required, but the lengths must describe a possible triangle. The triangle area from three sides rule uses side a (units), side b (units), side c (units) in Heron's formula; read each symbol as the field named on this page.

Interpreting another set of values

For triangle area from three sides, the fields labeled side a (units), side b (units), side c (units) are the known quantities. The calculator solves for area in square units using Heron's formula. Label the actual base, perpendicular height, side lengths, or right-angle legs before using the answer. A result for area cannot be compared directly with a side length because their units differ. When comparing two calculations, keep their units and input definitions identical.

For a second triangle area from three sides check, keep the other default inputs fixed and change side a from 3 units to 5 units. Area changes from 6 square units to 9.165151 square units. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

All three side lengths must be positive, and each must be shorter than the sum of the other two. A near-degenerate triangle can have a very small area; a combination that violates the triangle inequality has no real triangular area. Area is measured in square units. The reported area is an area. If the input lengths were centimeters, its unit would be square centimeters; the numerical 6 square units therefore describes covering rather than edge length.