To classify a triangle by its sides, call the longest side c and compare c2 with a2 + b2. If c2 < a2 + b2, the triangle is acute; if c2 = a2 + b2, it is right; if c2 > a2 + b2, it is obtuse. This is the Pythagorean inequality. Check first that the lengths form a triangle.
The three cases
An acute triangle has all three angles less than 90°. A right triangle has one angle of exactly 90°. An obtuse triangle has one angle greater than 90°. The largest angle always sits opposite the longest side, so comparing the square of the longest side with the other two squares tells you which kind you have:
| Compare | Largest angle | Triangle | Example |
|---|---|---|---|
| c2 < a2 + b2 | less than 90° | acute | 5, 7, 8: 64 < 25 + 49 = 74 |
| c2 = a2 + b2 | exactly 90° | right | 6, 8, 10: 100 = 36 + 64 |
| c2 > a2 + b2 | more than 90° | obtuse | 4, 6, 9: 81 > 16 + 36 = 52 |
The middle row is the converse of the Pythagorean theorem. The top and bottom rows extend it to every other triangle, which is why this result is sometimes called the Pythagorean inequality theorem.
Key ideaIf c2 is too small, the angle opposite c is squeezed shut: acute. If c2 is too big, the angle is stretched open: obtuse. Only when c2 is exactly a2 + b2 is the angle a right angle.
Step 1: check the triangle inequality
Not every set of three lengths makes a triangle. The triangle inequality says each side must be shorter than the other two added together. In practice you only need one check: the two shorter sides must add up to more than the longest side.
Skipping this step gives nonsense. For lengths 2, 3 and 6, the comparison 62 = 36 > 22 + 32 = 13 would say "obtuse", but there is no triangle to describe. If the two shorter sides add up to exactly the longest, as with 2, 3 and 5, the "triangle" collapses flat into a line segment. Geometers call that a degenerate triangle, and it doesn't count.
Step 2: compare the squares
- Sort the sides. Call the longest one c and the other two a and b.
- Check the triangle inequality: a + b > c.
- Square each side and add the two smaller squares: a2 + b2.
- Compare with c2 and name the triangle: smaller means acute, equal means right, larger means obtuse.
Common mistakesComparing the sides instead of their squares: a + b > c is true for every triangle, so it tells you nothing about angles. And using a side that is not the longest as c: for 9, 7, 5 the test 52 against 92 + 72 is meaningless, because the largest angle is opposite 9.
If you would rather check your answer with a tool, the right triangle checker runs both steps and shows the working.
Why the Pythagorean inequality works
Think of sides a and b as two rods joined by a hinge at C. When the hinge is nearly closed, the far ends are close together and c is short. As the hinge opens, c gets longer, steadily, all the way to 180°. Euclid proved this as Proposition 24 of Book I of the Elements, and today it is often called the hinge theorem: of two triangles with the same two sides, the one with the larger angle between them has the longer third side.
At exactly 90°, the Pythagorean theorem tells us c2 = a2 + b2. So any angle smaller than 90° gives a shorter c, and c2 < a2 + b2. Any angle larger than 90° gives a longer c, and c2 > a2 + b2. Because c only ever grows as the angle grows, the logic also runs backward: from c you can tell whether the angle is under, at or over 90°.
The law of cosines gives the same answer in one line. For any triangle,
c2 = a2 + b2 − 2abcosCThe last term is the "correction" to the Pythagorean theorem. The cosine of an acute angle is positive, so for acute C you subtract something and c2 comes out smaller than a2 + b2. At 90°, cosC = 0 and nothing changes. For an obtuse angle the cosine is negative, so subtracting it adds, and c2 comes out larger. Rearranged, a2 + b2 − c2 = 2abcosC: the sign of the gap between the two sides of the inequality is the sign of cosC.
Example: how obtuse is the 3-5-7 triangle?
The test says obtuse, because 49 > 9 + 25 = 34. The law of cosines gives the exact angle:
cosC= a2 + b2 − c2 over 2ab = 34 − 49 over 30 = −1 over 2C= 120°The largest angle is exactly 120°, the obtuse state shown in the animation above.
Worked examples
Example 1: 5, 7 and 8
Triangle check: 5 + 7 = 12 > 8. Then compare:
c2 = 82 = 64 a2 + b2 = 25 + 49 = 7464 < 74, so the triangle is acute. (Its largest angle is about 81.8°.)
Example 2: 4, 6 and 9
Triangle check: 4 + 6 = 10 > 9. Then:
c2 = 81 a2 + b2 = 16 + 36 = 5281 > 52, so the triangle is obtuse. (Its largest angle is about 127.2°.)
Example 3: a square root among the sides
Classify √7, 3 and 4. Since √7 ≈ 2.65, the longest side is 4, and 2.65 + 3 > 4.
c2 = 16 a2 + b2 = (√7)2 + 32 = 7 + 9 = 16Equal, so it is a right triangle. Squaring a square root simply removes the root.
Example 4: decimals
Classify 1.2, 1.6 and 2.1. Triangle check: 1.2 + 1.6 = 2.8 > 2.1.
c2 = 2.12 = 4.41 a2 + b2 = 1.44 + 2.56 = 44.41 > 4, so it is obtuse, but only just: the largest angle is about 96.1°. (With c = 2 it would be the right triangle 1.2-1.6-2.)
Example 5: two isosceles triangles
Compare 5, 5, 7 with 5, 5, 8. In both, a2 + b2 = 25 + 25 = 50.
72 = 49 < 50 82 = 64 > 505-5-7 is acute (barely) and 5-5-8 is obtuse. The right triangle sits between them, at c = √50 = 5√2 ≈ 7.07, the 45-45-90 triangle.
Practice: classify each set, or say that it is not a triangle. (a) 10, 11, 14 (b) 9, 40, 41 (c) 7, 7, 7 (d) 6, 7, 12 (e) 8, 9, 12 (f) 3, 4, 8 (g) 12, 35, 37 (h) 2, 2, 3.
Show answers
| Sides | a2 + b2 | c2 | Type |
|---|---|---|---|
| (a) 10, 11, 14 | 221 | 196 | Acute |
| (b) 9, 40, 41 | 1681 | 1681 | Right (a Pythagorean triple) |
| (c) 7, 7, 7 | 98 | 49 | Acute (every equilateral triangle is) |
| (d) 6, 7, 12 | 85 | 144 | Obtuse |
| (e) 8, 9, 12 | 145 | 144 | Acute, by a whisker (about 89.6°) |
| (f) 3, 4, 8 | — | — | Not a triangle: 3 + 4 < 8 |
| (g) 12, 35, 37 | 1369 | 1369 | Right |
| (h) 2, 2, 3 | 8 | 9 | Obtuse |
More sets to practice on are in the converse and classification worksheet.
Finding the range of the third side
A favorite test question turns the problem around: two sides are 6 and 8, so which lengths of the third side make the triangle acute? The trap is that the longest side changes depending on the answer.
- Triangle inequality. 8 − 6 < c < 8 + 6, so 2 < c < 14.
- When c is the longest side (c ≥ 8). Acute needs c2 < 36 + 64 = 100, so c < 10. Right at c = 10; obtuse for 10 < c < 14.
- When 8 is the longest side (c < 8). Now the angle opposite 8 is the one to watch. Acute needs 64 < 36 + c2, so c2 > 28 and c > √28 = 2√7 ≈ 5.29. Right at c = 2√7; obtuse for 2 < c < 2√7.
- Combine. The triangle is acute exactly when 2√7 < c < 10, that is, between about 5.29 and 10.
Questions like this appear on the SAT, ACT and in math competitions; the SAT and ACT guide has more.
Where can the third corner go?
Here is a picture that captures the whole Pythagorean inequality at once. Fix the side AB = c and ask where the third corner C can be. Draw the circle that has AB as its diameter.
Coordinates show why. Put A = (−r, 0) and B = (r, 0), so c = 2r, and let C = (x, y). Using the distance formula for the other two sides:
CA2 + CB2 − AB2= (x + r)2 + y2 + (x − r)2 + y2 − 4r2= 2(x2 + y2 − r2)So CA2 + CB2 is bigger than AB2 exactly when x2 + y2 > r2, which means C lies outside the circle x2 + y2 = r2. The Pythagorean inequality is the inside-or-outside test from the equation of a circle. One caution: outside the circle, angle C is acute, but the whole triangle is acute only if C also lies between the dashed lines; beyond them, the angle at A or B becomes obtuse instead.
Classifying triangles by sides and by angles
"Classify by sides" can mean two things, and side lengths settle both. Equal sides give the familiar names: scalene (no sides equal), isosceles (at least two equal) and equilateral (all three equal). The squares of the sides give the angle type. The two systems combine freely, with one exception:
| Acute | Right | Obtuse | |
|---|---|---|---|
| Scalene | 5, 7, 8 | 3, 4, 5 | 4, 6, 9 |
| Isosceles | 5, 5, 7 | 1, 1, √2 | 5, 5, 8 |
| Equilateral | any, e.g. 7, 7, 7 | impossible | impossible |
An equilateral triangle is always acute, because s2 < s2 + s2. And no triangle can have two angles of 90° or more, since its three angles add up to 180°. That is why testing the longest side alone classifies the whole triangle. Splitting an isosceles triangle down the middle is also the key to finding its height.
Frequently asked questions
What is the Pythagorean inequality theorem?
It extends the Pythagorean theorem to all triangles. With c the longest side, c2 < a2 + b2 means the triangle is acute and c2 > a2 + b2 means it is obtuse. Equality, c2 = a2 + b2, is the right-triangle case.
How can you tell if a triangle is acute, right or obtuse without measuring angles?
Square the three side lengths. Compare the largest square with the sum of the other two: smaller means acute, equal means right, bigger means obtuse. For 8, 9, 12 you get 144 against 64 + 81 = 145, so it is acute.
Why do I have to use the longest side?
The largest angle is always opposite the longest side, and it is the only angle that can be 90° or more. Testing a shorter side only tells you about a smaller angle, which is always acute, so the answer would be useless.
Do I need to check the triangle inequality first?
Yes. The squares comparison assumes a real triangle exists. Lengths 3, 4 and 8 give 64 > 25, which looks obtuse, yet 3 + 4 < 8, so the sides cannot even meet.
Can an isosceles triangle be obtuse?
Yes. In 5, 5, 8 the base is long enough that 64 > 25 + 25, so the angle at the top is obtuse (about 106.3°). An isosceles triangle can be acute, right or obtuse; only an equilateral triangle is forced to be acute.
How is the Pythagorean inequality related to the law of cosines?
The law of cosines, c2 = a2 + b2 − 2abcosC, contains it. The sign of cosC decides whether c2 falls below, on or above a2 + b2. The law of cosines page shows how to find the exact angle.
Keep exploring
- LearnConverse of the Pythagorean TheoremThe converse of the Pythagorean theorem explained: the statement, a clear proof, how to test any three side lengths, and many yes-or-no worked examples.
- LearnThe Law of CosinesThe law of cosines c² = a² + b² − 2ab·cos C explained: how it extends the Pythagorean theorem, a proof by dropping an altitude, and SAS and SSS examples.
- CalculatorsIs It a Right Triangle? Checker and Triangle ClassifierIs it a right triangle? Enter three side lengths to check the triangle inequality, classify the triangle as right, acute or obtuse, and find all three angles.
- PracticeConverse of the Pythagorean Theorem WorksheetConverse of the Pythagorean theorem worksheet: 16 printable problems on right, acute and obtuse triangles from their sides, with every step in the answer key.
- LearnEquation of a Circle and the Pythagorean TheoremWhy the equation of a circle, x² + y² = r², is the Pythagorean theorem, plus the center-radius form, worked examples and the inside/outside test.