Acute Triangle Calculator

Background.

Three side lengths fix a triangle completely, angles included, so you can tell whether it is acute, right or obtuse without ever picking up a protractor. The test needs only the longest side. Square it and compare the result with the squares of the other two added together: if the two shorter squares add up to more, every angle is under 90°; if they match exactly, the angle opposite the longest side is a right angle; if they fall short, that angle is obtuse.

The reason lies in the law of cosines, c² = a² + b² − 2ab·cos C. When C is exactly 90° its cosine is zero and the law collapses into Pythagoras' theorem. A smaller angle has a positive cosine, which makes c² less than a² + b²; a wider one has a negative cosine and pushes c² above it. Only the longest side needs checking because the largest angle always sits opposite it, and if the largest angle is under 90°, the other two must be as well.

The sides are also compared with each other. Three equal sides make the triangle equilateral, which is always acute; two equal make it isosceles; none equal make it scalene. The perimeter comes alongside.

Two practical points. Three lengths form a triangle only if the two shorter ones together exceed the longest — 2, 3 and 5 would lie flat — and such sets are refused. And the right-angle test is exact. A builder squaring a corner with a 3-4-5 rope gets a clean answer, but a diagonal measured as 1.41421356 on a unit square is a hair short of √2 and reads as acute, because real measurements almost never land precisely on right.

What is acute triangle calculator?

An acute triangle is one in which all three interior angles are less than 90°. A right triangle has exactly one 90° angle and an obtuse triangle has one angle wider than 90°. Because the angles of any triangle add up to 180°, at most one angle can reach or pass 90°, so the classification depends entirely on the largest angle, which lies opposite the longest side. Equilateral triangles are always acute; isosceles and scalene triangles can fall into any of the three classes.

How to use this calculator.

1. 01Enter the three side lengths in any order, all in the same unit — centimetres, feet or grid squares all work. The sides are sorted before testing, so it does not matter which one you call a, b or c.
2. 02Type lengths as precisely as you know them. The comparison is exact, so a triangle meant to be right-angled is reported as right only if a² + b² equals c² to the last digit entered.
3. 03If you get a triangle-inequality error, look for a typo: the two shorter sides must add up to more than the longest, or the three lengths cannot meet.
4. 04Read the classification, which pairs side type with angle type (for example isosceles acute triangle), and the perimeter in the same unit as your sides.

The formula.

Sort the sides so that c is the longest, then compare a² + b² with c². If a² + b² > c² the triangle is acute; if a² + b² = c² it is right; if a² + b² < c² it is obtuse. This follows from the law of cosines, c² = a² + b² − 2ab·cos C: the term 2ab·cos C is positive when the angle C opposite the longest side is under 90°, zero at 90° and negative beyond it. Since that angle is the largest in the triangle, testing it alone classifies all three. Before anything else, the two shorter sides must sum to more than the longest; if they only equal it, the three corners lie on one straight line. Side type comes from equality checks — all three sides equal, exactly two equal, or none — and the perimeter is a + b + c. Because no angle is ever computed, the right-angle case is detected exactly rather than through an approximate cosine.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.