Understand the triangle

A triangle is a plane polygon bounded by three straight sides. Its vertices determine three interior angles. A valid non-degenerate triangle has positive area, and the sum of its two shorter sides must exceed the longest side.

Properties that make the calculations work

In Euclidean geometry the interior angles total 180 degrees. Any side may be called the base; the matching height must be perpendicular to that base or its extension. Three side lengths determine a triangle up to reflection and rigid motion.

Where the model is useful

Use SSS when all side lengths are known, SAS when an included angle is measured, and base with height when only area is required. A base and height alone do not determine the perimeter.

Measure the right thing

The included angle in SAS lies between the two entered sides. A sloping side is usually not the perpendicular height.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k². This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Triangle provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.