How to Determine if a Triangle Is a Right Triangle
A right triangle has exactly one 90° interior angle. The most reliable way to test three known side lengths is the converse of the Pythagorean theorem: label the longest side c and the other two sides a and b, then check whether a² + b² = c². If the equation holds, the angle opposite c is exactly 90°. This calculator also validates that the three lengths can even form a triangle, classifies the triangle as acute, right, or obtuse, and reports the largest angle using the law of cosines.
How the calculation works
First, the calculator checks the triangle inequality — the two shorter sides must add up to more than the longest side (a + b > c), otherwise no triangle exists and the calculation stops there. Next, it sorts your three inputs so the longest is always treated as c, regardless of which field you typed it into, and computes a² + b² and c² for comparison. If the two sums are equal (within a small rounding tolerance to account for measured, decimal-rounded lengths), the triangle is a right triangle. If a² + b² is larger, every angle is under 90° and the triangle is acute; if a² + b² is smaller, the angle opposite c exceeds 90° and the triangle is obtuse. The exact size of that largest angle comes from the law of cosines, cos(C) = (a² + b² − c²) / (2ab), which reduces to the right-angle case (C = 90°) exactly when a² + b² = c².
Common mistakes
- Not identifying the longest side as c: the Pythagorean relationship only works when c is the longest side. Plugging in whichever side you entered third, instead of the actual longest side, gives a false negative.
- Expecting exact equality with measured lengths: real-world measurements are rounded, so a²+b² and c² may differ by a tiny amount even for a true right triangle. A small tolerance (built into this calculator) accounts for that rounding.
- Confusing "right triangle" with "isosceles right triangle": a right triangle only needs one 90° angle — it does not need two equal sides. Sides 3, 4, 5 form a right triangle with no equal sides at all.
Real-world applications
- Carpentry and construction use the "3-4-5 rule" (and its multiples, like 6-8-10) to square corners and foundations without a protractor
- Surveying and layout work verify that plots, rooms, or building footprints are truly rectangular by checking the diagonals
- Engineering and physics use right-triangle relationships constantly when resolving forces or vectors into perpendicular components
- Navigation and mapping use the Pythagorean theorem to find straight-line distances between points defined by perpendicular offsets