What Is The Converse Of The Pythagoras Theorem?
The converse of the Pythagoras theorem states: in a triangle, if the square of the longest side equals the sum of the squares of the other two sides, then the triangle is a right triangle, with the right angle opposite that longest side. In symbols, if the sides are $a$, $b$, $c$ with $c$ the longest, and
$$a^2 + b^2 = c^2$$
then the angle opposite $c$ is $90°$.
That is the whole statement. It runs the Pythagoras relationship backwards.
Here is the difference that matters, and the reason this topic is separate from the forward theorem. The Pythagoras theorem starts with a triangle you already know is right-angled and finds a missing side. The converse starts with three side lengths and tests whether the triangle is right-angled at all. One assumes the right angle and computes a length; the other assumes the lengths and deduces the angle.
Pythagoras theorem | Converse | |
|---|---|---|
You start with | a known right triangle | three side lengths |
You find | a missing side | whether the angle is $90°$ |
Direction | angle → side | sides → angle |
Relation used | $a^2 + b^2 = c^2$ | $a^2 + b^2 = c^2$ |
Proof Of the Converse
The converse is not obvious - knowing $a^2 + b^2 = c^2$ tells you about lengths, and the claim is about an angle. The proof bridges that gap by building a triangle we know is right-angled and showing the given one must match it.
Suppose triangle $ABC$ has sides $a$, $b$, $c$ with $a^2 + b^2 = c^2$, and side $c$ is opposite $\angle B$… let it be opposite the vertex we want to test.
Step 1. Construct a brand-new right triangle $PQR$ with a right angle at $Q$, legs $PQ = a$ and $QR = b$.
Step 2. By the forward Pythagoras theorem applied to $PQR$, its hypotenuse $PR$ satisfies
$$PR^2 = a^2 + b^2$$
Step 3. But we were given $a^2 + b^2 = c^2$, so
$$PR^2 = c^2 \implies PR = c$$
Step 4. Now compare triangles $ABC$ and $PQR$: all three sides match ($a = a$, $b = b$, and the third side $c = PR$). By the SSS (side-side-side) congruence rule, the two triangles are congruent.
Step 5. Congruent triangles have equal corresponding angles, so the angle in $ABC$ opposite $c$ equals the angle in $PQR$ opposite $PR$ - which is the $90°$ we built. The triangle is right-angled.
The Test Also Sorts Acute And Obtuse Triangles
The converse does more than spot right triangles. Compare $a^2 + b^2$ with $c^2$ (where $c$ is the longest side) and you can classify any triangle by its biggest angle:
If $a^2 + b^2 = c^2$ → right triangle (biggest angle exactly $90°$).
If $a^2 + b^2 > c^2$ → acute triangle (biggest angle less than $90°$).
If $a^2 + b^2 < c^2$ → obtuse triangle (biggest angle more than $90°$).
A memory hook: if the two short sides have "more than enough" squared length to match the longest side, the triangle is squeezed tight and stays acute; if they fall short, the longest side stretches the far corner open past $90°$.
Examples of the Converse of the Pythagoras Theorem
Example 1
A triangle has sides $6$, $8$, and $10$. Is it right-angled?
The longest side is $10$, so test $6^2 + 8^2$ against $10^2$.
$$6^2 + 8^2 = 36 + 64 = 100$$ $$10^2 = 100$$
The two are equal. Final answer: yes, it is a right triangle.
Example 2
A triangle has sides $8$, $10$, and $6$. A student squares them in the given order and writes $8^2 + 10^2 = 6^2$, gets $164 \ne 36$, and concludes it is not a right triangle. Correct?
The instinct is to plug the sides in the order they are written. But the converse only works when $c$ is the longest side - the formula tests the sum of the two shorter squares against the longest square.
Watch where it breaks: the student put $10$ (the longest side) on the left with $8$, and $6$ (a short side) alone on the right. That is not the correct comparison, so the "not right-angled" conclusion is unreliable.
The correct way sorts the sides first: shortest two are $6$ and $8$, longest is $10$.
$$6^2 + 8^2 = 36 + 64 = 100 = 10^2$$
They match. Final answer: it is a right triangle - the same one as Example 1, just listed in a different order.
Example 3
Classify the triangle with sides $6$, $8$, and $11$.
Longest side is $11$. Compare $6^2 + 8^2$ with $11^2$.
$$6^2 + 8^2 = 36 + 64 = 100$$ $$11^2 = 121$$
Since $100 < 121$, we have $a^2 + b^2 < c^2$. Final answer: an obtuse triangle.
Example 4
Classify the triangle with sides $4$, $5$, and $6$.
Longest side is $6$. Compare $4^2 + 5^2$ with $6^2$.
$$4^2 + 5^2 = 16 + 25 = 41$$ $$6^2 = 36$$
Since $41 > 36$, we have $a^2 + b^2 > c^2$. Final answer: an acute triangle.
Example 5
A triangle has sides $5$, $12$, and $13$. Show it is right-angled and name the hypotenuse.
Longest side is $13$. Test the converse.
$$5^2 + 12^2 = 25 + 144 = 169 = 13^2$$
The relation holds, so the triangle is right-angled, and the hypotenuse (the side opposite the right angle, always the longest) is $13$. This is a well-known Pythagorean triple. Final answer: right-angled, hypotenuse $= 13$.
Example 6
A carpenter marks a corner using lengths $9$, $12$, and $15$. Is the corner square?
The corner is between the two shorter sides; it is square only if the triangle is right-angled. Longest side is $15$.
$$9^2 + 12^2 = 81 + 144 = 225 = 15^2$$
The relation holds. Final answer: yes, the corner is a true right angle (a $3$-$4$-$5$ triple scaled by $3$).
Where the Converse Earns Its Keep: "Building a Right Angle Without a Protractor"
The forward theorem measures; the converse constructs. That is why the converse, not the theorem, is the one that shows up on building sites, in surveying, and in navigation.
Squaring a foundation. A builder with only a tape measure marks $3$ units along one edge, $4$ along the next, and checks the diagonal reads $5$. If it does, the corner is exactly $90°$ - no protractor needed. Scale it to $9$-$12$-$15$ for a bigger frame, as in Example 6.
Surveying and land plots. Right angles for property boundaries are set by measuring three lengths and checking the Pythagorean relation, because measuring an angle over long distances is far less accurate than measuring three lengths.
Classifying triangles fast. Give a computer three side lengths and the converse instantly labels the triangle right, acute, or obtuse with one comparison, no trigonometry required.
The Mistakes Students Make Most Often On Converse
Mistake 1: Not identifying the longest side first
Where it slips in: applying the converse to sides given in a random order.
Don't do this: square the three sides in the order they appear and test the first two against the third.
The correct way: the converse compares the two shortest squares against the longest square. The first instinct is to trust the given order, but the longest side must sit alone on one side of the comparison. Sort the sides before you square anything - that single habit prevents the most common wrong "not right-angled" answers, as Example 2 showed.
Mistake 2: Confusing the converse with the theorem
Where it slips in: a problem gives three sides and asks whether the triangle is right-angled.
Don't do this: try to "find the hypotenuse" as if the right angle were already known.
The correct way: when the angle is unknown and the sides are known, you are using the converse to test, not the theorem to compute. The point of confusion between the two directions is real - the theorem assumes the right angle, the converse proves it. Ask first: am I given the angle, or checking for it?
Mistake 3: Forgetting the acute and obtuse cases
Where it slips in: classifying a triangle that turns out not to be right-angled.
Don't do this: stop at "$a^2 + b^2 \ne c^2$, so not right" and give no further classification.
The correct way: the inequality still tells you the type - greater means acute, less means obtuse.
Conclusion
The converse of the Pythagoras theorem says that if $a^2 + b^2 = c^2$ (with $c$ the longest side), the triangle is right-angled.
It reverses the theorem: the theorem finds a side from a known right angle; the converse deduces the angle from known sides.
The proof builds a right triangle with the same sides and matches the two by SSS congruence.
The same comparison classifies any triangle: equal means right, $a^2 + b^2 > c^2$ means acute, and $a^2 + b^2 < c^2$ means obtuse.
To take the converse of the Pythagoras theorem further with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor, or math classes online.
A Practical Next Step
Practice these to solidify your understanding: test whether $7$, $24$, $25$ is a right triangle; classify the triangle with sides $9$, $10$, $12$ as right, acute, or obtuse; and explain why a builder uses $3$-$4$-$5$ rather than a protractor. If ordering the sides trips you, return to Example 2 and sort before you square. Want a live Bhanzu trainer to walk through more triangle problems? Book a free demo class.
Read More
Properties of a triangle - the angle and side rules every triangle obeys.
Congruence in triangles - the SSS rule the converse proof relies on.
Angles of a rectangle - where the 3-4-5 squaring check keeps corners at 90°.
Acute triangle - one of the triangle types the converse test identifies.
Right-angled triangle - the triangle the converse confirms from three side lengths.
Was this article helpful?
Your feedback helps us write better content