Triangle Area Calculator

Calculate triangle area from three sides using Heron's formula

What is Heron's Formula Triangle Calculator?

Heron's formula triangle calculator is a mathematical tool that determines the area of any triangle when only the lengths of its three sides are known. Named after Hero of Alexandria (circa 10-70 AD), this formula eliminates the need to know angles or heights, making it invaluable for surveying, construction, and engineering applications where direct measurement of area is impractical. The formula uses the semi-perimeter and all three side lengths to calculate the exact area.

Our comprehensive calculator not only computes the area using Heron's formula but also validates triangle inequality, determines triangle type (scalene, isosceles, equilateral), calculates perimeter, and provides detailed geometric analysis. It's essential for land surveyors measuring irregular plots, construction professionals calculating material needs, architects designing triangular spaces, and students learning geometry. The tool includes triangle validity checking and comprehensive error handling.

This calculator is indispensable for surveyors measuring property boundaries, construction estimators calculating triangular areas, engineers designing structural elements, navigation specialists using triangulation, and educators teaching geometric principles. Accurate area calculations ensure proper planning, material estimation, and successful project completion across various technical fields.

Three Side Lengths

Input triangle sides a, b, c

Heron's Formula

Mathematical calculation

Triangle Properties

Area and geometric data

Triangle Side Lengths

Side a Side b Side c
Enter length of side A
Enter length of side B
Enter length of side C
Area
0
units²
Triangle area calculated using Heron's formula.
Perimeter
0
units
Sum of all three triangle sides.
Triangle Type
-
Classification based on side lengths.

Common Triangle Examples

3-4-5 Triangle
Right triangle (Area: 6)
5-5-5 Triangle
Equilateral (Area: 10.83)
5-5-8 Triangle
Isosceles (Area: 12)
6-8-10 Triangle
Right triangle (Area: 24)
7-9-11 Triangle
Scalene (Area: 30.07)
13-14-15 Triangle
Scalene (Area: 84)

Heron's Formula & Triangle Mathematics

Heron's Formula

Area Calculation:
A = s(s − a)(s − b)(s − c)

Where s is the semi-perimeter and a, b, c are the side lengths. Named after Hero of Alexandria.

Wolfram - Heron's Formula
Semi-perimeter:
s = a + b + c2

Half the perimeter of the triangle. Essential component of Heron's formula calculation.

Triangle Inequality:
a + b > c
a + c > b
b + c > a

Conditions that must be satisfied for three lengths to form a valid triangle.

Triangle Classification

By Side Lengths:
Equilateral: a = b = c
Isosceles: a = b ≠ c
Scalene: a ≠ b ≠ c

Classification based on equality of side lengths. Affects geometric properties and calculations.

Khan Academy - Triangle Types
Right Triangle Test:
a2 + b2 = c2

Pythagorean theorem test for right triangles. If satisfied, triangle has a 90° angle.

Mathematical References

Hero of Alexandria (10-70 AD)

Ancient Greek mathematician who developed the formula for triangle area from side lengths.

Mathematical Biography
Wolfram MathWorld

Comprehensive mathematical reference for Heron's formula and triangle geometry.

Heron's Formula Reference
National Council of Teachers of Mathematics

Educational standards and teaching resources for triangle geometry and area calculations.

NCTM Standards

Calculation Methodology

Numerical Stability:

Formula implementation uses careful ordering and precision handling to avoid floating-point errors.

Validation Process:

Triangle inequality theorem verified before area calculation to ensure valid triangle geometry.

Precision Considerations:

Results accurate to machine precision (~15 decimal places) using IEEE 754 double-precision arithmetic.

Applications:

Used in surveying, navigation, computer graphics, engineering design, and geometric analysis.