What's the triangle inequality?

For any triangle: the sum of any two sides must exceed the third. Equivalently, the longest side < sum of the other two. If a=3, b=4, c=10, no triangle exists (3+4 < 10).

Hero of Alexandria, "Metrica" (~60 AD) — original derivation of Heron's formula. Euclid, "Elements" Book I Proposition 41 — base × height triangle area. Modern compilation: Coxeter HSM (1969) "Introduction to Geometry" 2nd ed.

What's the triangle inequality?

For any triangle: the sum of any two sides must exceed the third. Equivalently, the longest side < sum of the other two. If a=3, b=4, c=10, no triangle exists (3+4 < 10).

Hero of Alexandria, "Metrica" (~60 AD) — original derivation of Heron's formula. Euclid, "Elements" Book I Proposition 41 — base × height triangle area. Modern compilation: Coxeter HSM (1969) "Introduction to Geometry" 2nd ed.

Triangle Area Calculator (3 methods)

Base × height, Heron's formula (SSS), or side-angle-side (SAS) — pick the inputs you have.

Result

Triangle area

30.0000

Method: base-height.

Methodbase-height
Base10.0000
Height6.0000
Area30.000000

Step-by-step

Area = ½ · base · height = ½ · 10 · 6 = 30.0000.

How to use this calculator

Pick the method that matches your inputs.
Base+height: orthogonal height to the base (drop a perpendicular if needed).
SSS (Heron): three side lengths.
SAS: two sides and the angle BETWEEN them (not opposite).

About this calculator

Triangle area has three classic formulas depending on what you can measure. Base × height (½·b·h) is the textbook formula but requires knowing the perpendicular height. Heron's formula (named after Hero of Alexandria, ~60 AD) gives the area from just the three side lengths via the semi-perimeter: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. Side-Angle-Side (SAS) uses two sides and the included angle: A = ½·a·b·sin(C) — useful in surveying and trigonometry where one angle is known. All three give identical results for a valid triangle; pick the one whose inputs match your situation. Triangle inequality check: the longest side must be less than the sum of the other two — the tool flags violations explicitly.

Frequently asked

It requires no trigonometry and no perpendicular measurement — just the three side lengths. Surveyors, builders, and astronomers used it for ~1900 years before modern trig calculators.