Acute Triangles Explained: the a² + b² > c² rule, with worked examples

What an acute triangle actually is

An acute triangle is a triangle whose three interior angles each measure strictly less than 90°. That is the entire definition — no mention of side lengths, area, or symmetry. Because the three angles of any triangle always add to exactly 180°, this single constraint is enough to pin the shape down to a recognisable family: every angle sits in the open interval (0°, 90°), the largest one in particular, and none of the three is allowed to touch the right angle. The acute triangle calculator does the classification automatically — type three side lengths and it tells you whether what you have is acute, right, or obtuse, along with the actual angles and area.

It is worth stressing the word strictly. A triangle with angles 90°, 45°, 45° is a right triangle, not an acute one, even though two of its angles are below 90°. The three classes — acute, right, obtuse — are mutually exclusive and they partition all possible triangles. Every triangle on the plane falls into exactly one of those three buckets, and the bucket is decided by whichever single angle happens to be the largest.

The shortcut: a² + b² versus c²

You can decide whether a triangle is acute without computing any angles at all, just by squaring the side lengths. Sort the sides so that c is the longest, and compare a² + b² with c²:

• a² + b² > c² → triangle is acute.
• a² + b² = c² → triangle is right (the Pythagorean equality).
• a² + b² < c² → triangle is obtuse.

This is sometimes called the converse, or sharpened, Pythagorean rule. It is not a coincidence — it falls out of the law of cosines the moment you write it down. The law of cosines says that c² = a² + b² − 2ab·cos(C), where C is the angle opposite the longest side. Rearranging gives 2ab·cos(C) = a² + b² − c², so the sign of cos(C) matches the sign of a² + b² − c². Since the cosine function is positive for angles below 90°, zero at exactly 90°, and negative above 90°, the comparison of a² + b² with c² reads off the sign of cos(C), which reads off the side of 90° on which the largest angle sits. The acute triangle calculator applies exactly this test, with one caveat: it also runs the triangle inequality check first, to rule out side triples that don’t form a triangle in the first place.

The reason you only have to check the largest angle is that the largest angle always sits opposite the longest side, and the other two angles must each be smaller. If the largest is below 90°, the other two are too, and you are done. If the largest is at or above 90°, the triangle is already disqualified — no amount of checking the smaller angles can rescue it.

Worked example: sides 5, 6, 7

Take the canonical small example: a triangle with sides 5, 6, and 7. Step by step:

1. Triangle inequality. 5 + 6 = 11 > 7, 5 + 7 = 12 > 6, 6 + 7 = 13 > 5. Every side is strictly less than the sum of the other two, so a triangle exists.
2. Identify the longest side. c = 7, so the largest angle is C (opposite c).
3. Sharpened Pythagorean test. a² + b² = 25 + 36 = 61. c² = 49. 61 > 49, so the triangle is acute. Done — if you only care about the classification, the test stops here.
4. Angles, via the law of cosines. cos(C) = (a² + b² − c²) / (2ab) = (25 + 36 − 49) / (2·5·6) = 12 / 60 = 0.2, so C = arccos(0.2) ≈ 78.46°. Similarly cos(A) = (b² + c² − a²) / (2bc) = (36 + 49 − 25) / (2·6·7) = 60 / 84 ≈ 0.7143, so A ≈ 44.42°. The remaining angle is B = 180° − A − C ≈ 57.12°. All three are below 90° — the test was right.
5. Area, via Heron’s formula. Semi-perimeter s = (5 + 6 + 7) / 2 = 9. Area = √(9·(9 − 5)·(9 − 6)·(9 − 7)) = √(9·4·3·2) = √216 ≈ 14.697 square units.

Punch the same numbers into the acute triangle calculator and you should see exactly these results: classification "Acute ✓", angles 44.42° / 57.12° / 78.46°, area 14.697.

The geometry behind the inequality

The sharpened Pythagorean rule has a satisfyingly visual interpretation. Imagine fixing two sides a and b at a vertex and slowly opening the angle C between them. The opposite side c grows as C grows. At C = 90° you have a right triangle and c² = a² + b² exactly — that is Pythagoras. As C opens past 90°, c keeps growing and c² overshoots a² + b²; as C closes inside 90°, c shrinks and c² falls below a² + b². The comparison between a² + b² and c² is literally measuring how far C is from 90°, in squared-length units.

The same story applies symmetrically to the other two angles — but you only ever need to check the largest, because if the largest passes the test the smaller two automatically do.

Acute, right, obtuse: the three regions

In the space of all possible triangles you can parametrise by the three side lengths (modulo scaling), the acute, right, and obtuse families form three regions separated by a single boundary surface — the right-triangle surface a² + b² = c². Move into the region where a² + b² > c² and you are looking at an acute triangle; move across to a² + b² < c² and the triangle is obtuse. The boundary itself is the set of right triangles, which is "measure zero": a random triangle picked from any sensible continuous distribution over side lengths will be either acute or obtuse with probability 1, and right with probability 0. Right triangles are special because we construct them deliberately, not because they show up often by accident.

Properties that follow from being acute

All three altitudes land inside the triangle

An altitude is the perpendicular dropped from one vertex onto the opposite side. In an acute triangle, every altitude has its foot somewhere on the opposite side itself — not on the extension. That is a clean diagnostic: if you ever drop an altitude and the foot falls outside the segment, you are looking at an obtuse triangle. Right triangles are the boundary case where two of the altitudes coincide with the legs.

The circumcentre sits inside the triangle

The circumcentre is the common intersection of the three perpendicular bisectors of the sides and the centre of the circle that passes through all three vertices. In an acute triangle, the circumcentre is strictly inside the triangle. In a right triangle, it lies exactly on the midpoint of the hypotenuse. In an obtuse triangle, it sits outside the triangle entirely. Likewise the orthocentre (the intersection of the three altitudes) is inside an acute triangle, at the right-angle vertex of a right triangle, and outside an obtuse one.

Squaring trick for area sanity

For an acute triangle with sides a, b, c, you can sanity-check Heron’s formula against the SAS expression. If C is the angle opposite the longest side c, then area = ½·a·b·sin(C). For the 5-6-7 triangle above, sin(78.46°) ≈ 0.9798, and ½·5·6·0.9798 ≈ 14.697 — exactly the Heron answer. The two formulas always agree; if they disagree numerically by more than rounding noise, one of the inputs is wrong.

How to think about borderline cases

Almost-right triangles

A triangle with sides 3, 4, 5.0001 is technically obtuse — c² is a hair above a² + b² — but visually indistinguishable from the classic 3-4-5 right triangle. In floating-point arithmetic these borderlines are sensitive: the acute triangle calculator uses a small tolerance (10⁻⁹) when comparing the largest angle with 90°, so a measured triangle that is "right within the noise floor of the measurement" is classified as right rather than forced into acute or obtuse by a rounding artefact. If you need more precision than that — say, because the inputs are exact rational numbers and you want an exact classification — do the sharpened Pythagorean test by hand on the squared integers and trust the integer comparison over the floating-point arccos.

Long, thin triangles

An acute triangle can be surprisingly thin. Sides 100, 100, 1 give an isoceles triangle with apex angle below 1° and base angles each just below 90° — still acute, just barely. Sides 1, 1, 1.999 give an isoceles triangle with base angles each near 2° and an apex angle near 176° — distinctly obtuse. The sharpened Pythagorean rule cleanly separates the two: for the first, 100² + 1² = 10001 > 100² = 10000; for the second, 1² + 1² = 2 is much less than 1.999² ≈ 3.996.

The boundary itself

Pythagorean triples — (3, 4, 5), (5, 12, 13), (8, 15, 17), and so on — are the integer-sided right triangles. Bumping the longest side down by any positive amount turns each of them acute; bumping it up by any positive amount turns it obtuse. That is a useful intuition pump: if you start from a right triangle and shorten the hypotenuse while keeping the legs fixed, you are forcing the opposite angle to close past 90° and become acute.

How to avoid the common mistakes

• Forgetting to identify the longest side first. The a² + b² > c² rule only works if c is the longest. If you label the sides in some other order — by which vertex you met first, say — the comparison will give nonsense answers. Always sort the sides before applying the test.
• Confusing "acute" with "small angles everywhere". An acute triangle can have one angle very close to 90° (think 89.99°). What it cannot have is any angle at or above 90°.
• Skipping the triangle inequality. Three positive numbers don’t always form a triangle. Sides 1, 2, 5 don’t — 1 + 2 = 3 is less than 5. The Calc Dragon acute triangle calculator checks this first and refuses to classify if the inequality fails. If you’re doing the arithmetic by hand, do the same.
• Comparing the wrong sides squared. "a² + b² vs c²" is a test about side lengths, not about which is algebraically called a or b or c. Reorder so that c is the longest and you avoid this trap entirely.
• Trusting an obviously-degenerate calculator output. If the calculator returns an angle of exactly 0° or exactly 180°, the triangle has collapsed onto a line and isn’t really a triangle. That usually means your inputs are right at the edge of the triangle inequality.

Where this comes up in practice

Classifying triangles as acute, right, or obtuse comes up in introductory geometry classes and in any field that meshes polygons together. In computer graphics, mesh quality often means "no obtuse triangles" because obtuse triangles produce long, sliver-like cells that destabilise numerical solvers built on top of them — Delaunay triangulation is in part a maximise-the-minimum-angle algorithm that tries to keep triangles as close to equilateral as possible, which is the most acute triangle there is. In structural and survey contexts, the same sharpened Pythagorean test is the cheapest way to confirm a nominally right angle is actually right and not "right within the survey tolerance".

For deeper exploration of any of the underlying formulas, the law of cosines calculator gives you cos(C) from the sides directly, the law of sines calculator relates the three side-angle pairs, the Pythagorean theorem calculator handles right triangles specifically, and the Heron’s formula calculator gives area from three sides without going via angles. For a general SSS / SAS / ASA / AAS solver — not just acute triangles — see the general triangle calculator.

When the classification isn’t enough

Telling an acute triangle from a right or obtuse one is almost always the easy part. The harder questions in real geometry are things like "is this surveyed plot of land actually a rectangle or does it have a small angle defect that matters?" or "if I machine this part with these tolerances, will the corner sit in the acceptable range?" Those are not classification questions; they are tolerance questions, and they need you to think explicitly about how measurement error propagates through the sharpened Pythagorean comparison. If your largest angle comes out at 89.0° and your side-length measurements are good to ±1%, you cannot conclude the triangle is genuinely acute — the true angle could easily be above 90°. The classifier the acute triangle calculator gives you is exact for the numbers you typed; it is up to you to know how much those numbers can be trusted.

Frequently asked questions

See the FAQ section below the calculator on the acute triangle calculator page for direct answers to the most common questions — how to test acuteness from sides alone, whether equilateral and isoceles triangles can be acute, why the calculator sometimes refuses inputs, and how the area formula relates to Heron’s. The same FAQ items appear here marked up with FAQPage schema so search engines can surface them directly.