How to Tell if a Triangle Is Right, Acute or Obtuse From Its Sides

Geometry Lessons

The short answer

compare a² + b² against c²

Sort the sides so c is the longest. Equal means right angled, greater means acute, and smaller means obtuse. One comparison settles it.

This is Pythagoras used as a test rather than as a calculation. Instead of solving for a missing side, you check whether the relationship holds, and how it fails when it does not tells you the shape of the triangle without computing a single angle.

Why the comparison works

Picture the two shorter sides hinged at a corner, with the longest side closing the gap between their far ends. Open that hinge wider and the gap grows. Close it and the gap shrinks. The longest side is a direct measure of how open that corner is.

At exactly 90 degrees the relationship is Pythagoras, so the squares balance. Open the corner past 90 and the opposite side has to stretch further than Pythagoras allows, so its square overshoots. Close it below 90 and the opposite side falls short, so its square comes in under. The direction of the imbalance is the classification.

1

Sort and take the longest

Only the largest angle can be right or obtuse, and it always faces the longest side.

2

Square the two shorter ones and add

This is the value Pythagoras would predict for the longest side squared.

3

Compare against the longest squared

Equal is right angled, larger is acute, smaller is obtuse.

Three worked examples

Sides of 3, 4 and 5. The two shorter squares are 9 and 16, adding to 25, and 5 squared is also 25. They balance exactly, so the triangle is right angled.

Sides of 4, 5 and 6. The shorter squares are 16 and 25, adding to 41, while 6 squared is 36. The sum overshoots the longest, so the largest corner is tighter than 90 degrees and every angle is acute. It works out to 82.82 degrees.

Sides of 5, 6 and 10. The shorter squares are 25 and 36, adding to 61, while 10 squared is 100. The sum falls well short, so the largest angle is obtuse, and it comes to 130.54 degrees.

Three panels comparing the longest side squared against the sum of the other two squared. Three four five gives twenty five equals twenty five and is right angled, four five six gives forty one against thirty six and is acute, and five six ten gives sixty one against one hundred and is obtuse.
The circled corner in each panel is the one being classified. Because only one angle in a triangle can reach or exceed 90 degrees, testing that one settles the whole shape.

Sort before you compare, not after

Comparing a pair that does not include the longest side gives an answer about the wrong corner. A triangle with sides 5, 6 and 10 looks acute if you test the 5 and the 6 against each other, and it is actually obtuse by a wide margin.

test 5 and 10 against 6gives125 against 36wrong corner

this classifies a corner that was never in doubt

test 5 and 6 against 10gives61 against 100obtuse

the longest side is the only one that decides the shape

Only one angle can be the problem

The reason a single comparison is enough comes from the angle sum. Two angles of 90 degrees would already use up the full 180, leaving nothing for the third, so no triangle can hold more than one right or obtuse angle. Test the largest and everything else is settled by elimination.

It also means the three categories are exclusive rather than overlapping. A triangle is right angled, or acute, or obtuse, and never two of them.

Why the exact case is worth trusting

When the squares balance exactly, the angle is exactly 90 degrees and not merely close to it. That is worth knowing when the sides came from measurement, because 6, 8 and 10.01 is not a right triangle, and whether that matters depends on your tolerance rather than on the arithmetic.

Field measurements almost never balance exactly. The useful question there is how far off the balance sits relative to the precision you were working to, not whether it landed on zero.

Mistakes that catch people out

  • Comparing without sorting first. The test only means something when the longest side is on its own side of the comparison.
  • Reading the inequality backwards. A sum larger than the longest square means acute, not obtuse. Larger sum equals tighter corner.
  • Adding before squaring. Square each side first. 3 plus 4 squared is not 25.
  • Skipping the triangle inequality. Lengths that cannot form a triangle will still produce a comparison, and it means nothing.
  • Expecting field measurements to balance exactly. They rarely do. Decide what tolerance counts as square before you measure.

Practice questions

  1. Sides of 6, 8 and 10. Classify the triangle.
  2. Sides of 7, 8 and 12. Classify the triangle.
  3. Sides of 9, 12 and 15. Classify the triangle.
  4. Sides of 2, 3 and 4. Classify the triangle.
  5. Sides of 5, 5 and 7. Classify the triangle.
Show the answers
  1. 36 + 64 = 100 and 10² = 100, so right angled
  2. 49 + 64 = 113 and 12² = 144, so obtuse
  3. 81 + 144 = 225 and 15² = 225, so right angled
  4. 4 + 9 = 13 and 4² = 16, so obtuse
  5. 25 + 25 = 50 and 7² = 49, so acute, but only just

Common questions

How do I tell if a triangle is right, acute or obtuse from its sides?

Sort the sides, then compare the sum of the two shorter squares against the longest squared. Equal means right angled, greater means acute, and smaller means obtuse.

Which side do I compare against?

Always the longest. The largest angle faces it, and that is the only angle that can be right or obtuse.

Can a triangle have two obtuse angles?

No. Two angles above 90 degrees would already exceed 180 between them, leaving nothing for the third. At most one angle can be obtuse.

Do I need to calculate any angles?

No. The comparison classifies the triangle on its own. Calculate the angle only if you need its actual value.

What if the sides cannot form a triangle at all?

Test that first with the triangle inequality. If the two shorter sides do not add to more than the longest, there is no shape to classify.

How does this relate to the cosine rule?

It is the sign of the cosine. A positive cosine means acute, zero means right angled and negative means obtuse, which is exactly what this comparison detects without the division.

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