The converse of the Pythagorean theorem says that if the three sides of a triangle satisfy a2 + b2 = c2, where c is the longest side, then the triangle is a right triangle, and the right angle is the one opposite c. It lets you prove a triangle is right from its side lengths alone, without a protractor.
The theorem and its converse
Every if-then statement has a converse, made by swapping the "if" part and the "then" part. The Pythagorean theorem and the Pythagorean theorem converse are such a pair. In both, a, b and c are the side lengths, c is the longest, and C is the corner opposite side c.
The theorem
If a triangle has a right angle at C, then a2 + b2 = c2.
Use it to find a missing side of a triangle you already know is right.
The converse
If a triangle's sides satisfy a2 + b2 = c2, then it has a right angle at C.
Use it to decide whether a triangle is right when you know all three sides.
A converse is not automatically true just because the original statement is. "If an animal is a dog, it has four legs" is true, but its converse, "if an animal has four legs, it is a dog," is false. So the converse of the Pythagorean theorem needs a proof of its own. It has one: Euclid proved it as Proposition 48 of Book I of the Elements, directly after proving the theorem itself as Proposition 47.
Because both directions are true, they combine into a single "if and only if" statement:
∠C = 90° ⇔ a2 + b2 = c2Key ideaThe squares of the two shorter sides add up to the square of the longest side exactly when the triangle has a right angle. Every right triangle passes this test, and no other triangle does.
Why the converse is true: a proof
The standard proof builds a second triangle that we know is right and shows that it is an exact copy of the first. It needs the Pythagorean theorem and one congruence rule, SSS: if the three sides of one triangle equal the three sides of another, the triangles are congruent, so their angles match too.
- Start with what you are given. Triangle ABC has sides BC = a, CA = b and AB = c, and a2 + b2 = c2. Nothing is known yet about angle C.
- Build a right triangle with the same legs. Draw a right angle at a new point F. Measure FE = a along one arm and FD = b along the other, then join D to E.
- Find its hypotenuse. Triangle DEF is right, so the Pythagorean theorem applies: DE2 = a2 + b2. We were told that a2 + b2 = c2, so DE2 = c2. Lengths are positive, so DE = c.
- Compare the two triangles. Both have sides of length a, b and c. By SSS, △ABC ≅ △DEF, with C matching F because each is the corner between the sides of length a and b.
- Conclude. Matching angles of congruent triangles are equal, so ∠C = ∠F = 90°. Triangle ABC is a right triangle, which is what we wanted to prove.
Notice that step 3 uses the theorem itself. That is typical: the converse is proved from the theorem, which is why textbooks teach the theorem first. You can read Euclid's own version, which uses the same idea, in Elements Book I, Proposition 48, and his proof of the theorem on the Euclid's proof page.
There is a second, shorter route through the law of cosines, c2 = a2 + b2 − 2abcosC. If c2 = a2 + b2, the last term must be zero, so cosC = 0. The only angle between 0° and 180° whose cosine is zero is 90°. It is quick, but it depends on trigonometry, which is itself usually built on the Pythagorean theorem.
See it with a hinge
Picture two sticks, 3 and 4 units long, joined by a hinge at C. As the hinge opens, the gap between the free ends, side c, gets longer. There is exactly one opening where c2 equals 32 + 42 = 25, and that opening is 90°.
The same picture explains a failed test. If c2 is smaller than a2 + b2, the angle is narrower than 90° and the triangle is acute; if c2 is larger, the angle is wider and the triangle is obtuse. The full classification is on the acute, right or obtuse triangles page. Because c grows steadily as the hinge opens, each length of c matches exactly one angle. That is another way to see why the converse must hold.
How to tell if a triangle is a right triangle
Here is the test, step by step. It works for whole numbers, decimals, fractions and square roots.
- Check that the lengths make a triangle. The two shorter sides must add up to more than the longest. Lengths 2, 3 and 7 fail, because 2 + 3 < 7, so there is no triangle to test.
- Call the longest side c. It is the only possible hypotenuse. The other two are a and b, in either order.
- Square all three lengths. Work with exact values. Squaring a square root just removes the root: (√7)2 = 7.
- Compare a2 + b2 with c2. If they are equal, the triangle is right, and the right angle is opposite c. If they are not equal, the triangle is not right.
Common mistakeUsing the sides in the order they are listed. For lengths 13, 5 and 12, the test 132 + 52 = 122 gives 194 ≠ 144 and a wrong "no." The longest side is 13, so the correct test is 52 + 122 = 25 + 144 = 169 = 132. It is a right triangle.
TipIf all three lengths share a factor, divide it out first. For 15, 36 and 39, divide by 3 to get 5, 12 and 13, a known 5-12-13 right triangle. Scaling never changes the answer, because (ka)2 + (kb)2 = k2(a2 + b2).
Worked examples: right triangle or not?
Example 1: whole numbers, yes
Do 9, 12 and 15 form a right triangle? The longest side is 15.
92 + 122= 81 + 144 = 225152= 225Equal, so yes. The right angle is opposite the side of length 15. (These are the sides of a 3-4-5 triangle multiplied by 3.)
Example 2: whole numbers, no
Do 7, 8 and 11 form a right triangle?
72 + 82= 49 + 64 = 113112= 121113 ≠ 121, so no. Because c2 is the bigger number, the largest angle is wider than 90°: the triangle is obtuse.
Example 3: decimals
Is a triangle with sides 1.5 m, 2 m and 2.5 m right?
1.52 + 22= 2.25 + 4 = 6.252.52= 6.25Yes. Halve 3, 4 and 5 and you get exactly these lengths.
Example 4: square roots
Test √5, 2 and 3. Since √5 ≈ 2.24, the longest side is 3.
(√5)2 + 22= 5 + 4 = 932= 9Yes, it is a right triangle. Now try √2, √3 and 2: 2 + 3 = 5 but 22 = 4, so that one is not right (it is acute). Need a refresher on roots? See square roots and radicals.
Example 5: fractions
Test 1 over 4, 1 over 3 and 5 over 12. The longest is 5 over 12.
(1 over 4)2 + (1 over 3)2= 1 over 16 + 1 over 9 = 9 over 144 + 16 over 144 = 25 over 144(5 over 12)2= 25 over 144Yes. Multiply every side by 12 and you get 3, 4 and 5 again.
Example 6: a near miss
A student measures a triangle as 8 cm, 15 cm and 17.1 cm.
82 + 152= 64 + 225 = 28917.12= 292.41Not equal, so by the numbers it is not a right triangle, though it is very close to one (8-15-17 is right). With measured lengths, "almost equal" means "almost a right angle"; with exact lengths, the two sides must match exactly.
Try these yourself before opening the answers: (a) 6, 8, 10 (b) 4, 5, 6 (c) 8, 15, 17 (d) 12, 16, 21 (e) 1, 1, √2 (f) 11, 60, 61 (g) 0.9, 1.2, 1.5 (h) 3, 3, 4 (i) 1, √3, 2 (j) 6, 7, 9.
Show answers
| Sides | a2 + b2 | c2 | Right triangle? |
|---|---|---|---|
| (a) 6, 8, 10 | 100 | 100 | Yes |
| (b) 4, 5, 6 | 41 | 36 | No (acute) |
| (c) 8, 15, 17 | 289 | 289 | Yes |
| (d) 12, 16, 21 | 400 | 441 | No (obtuse) |
| (e) 1, 1, √2 | 2 | 2 | Yes (a 45-45-90 triangle) |
| (f) 11, 60, 61 | 3721 | 3721 | Yes |
| (g) 0.9, 1.2, 1.5 | 2.25 | 2.25 | Yes |
| (h) 3, 3, 4 | 18 | 16 | No (acute) |
| (i) 1, √3, 2 | 4 | 4 | Yes (a 30-60-90 triangle) |
| (j) 6, 7, 9 | 85 | 81 | No (acute) |
For a full printable set, use the converse of the Pythagorean theorem worksheet. To check your own numbers instantly, with the working shown, use the right triangle checker.
The carpenter's 3-4-5 check
Builders use the converse every day to make sure a corner is square, meaning exactly 90°. Laying out a deck, a foundation, a garden bed or a tiled floor, they don't trust their eyes or a small plastic square over long distances. They measure.
- Mark 3 units along one edge, measured from the corner.
- Mark 4 units along the other edge.
- Measure between the marks. If the distance is exactly 5 units, the corner is square, because 32 + 42 = 52. If it is more than 5, the corner is open too wide; if less, it is too tight. Adjust and measure again.
Bigger triangles give better accuracy. With a 3-4-5 ft triangle, a diagonal that is 1 over 8 inch too long means the corner is about 90.25°. With 9-12-15 ft, the same 1 over 8-inch error means only about 90.08°, three times closer. That is why builders scale up to 6-8-10 or 9-12-15, or in metric to 0.6-0.8-1 m or 3-4-5 m. Any Pythagorean triple works, including 5-12-13.
Example 7: squaring a patio
A rectangular patio is to measure 12 ft by 16 ft. What should the diagonal measure if the corners are square?
d2= 122 + 162 = 144 + 256 = 400d= √400 = 2020 ft. If the tape reads exactly 20 ft from corner to corner, that corner is 90°. More layout tricks are on the Pythagorean theorem in construction page.
You may read that ancient Egyptian surveyors laid out right angles with a rope tied into 12 equal sections, forming a 3-4-5 triangle. It is a popular story, but there is little direct evidence for it; the Egyptian rope stretchers page separates what is known from what is guessed.
Theorem or converse: which do you need?
Students often mix the two up. Ask yourself what you already know and what you are trying to find:
| You know | You want | Use |
|---|---|---|
| The triangle is right, plus two sides | The third side | The theorem |
| All three sides | Whether it is a right triangle | The converse |
| All three sides, and it is not right | Acute or obtuse | The Pythagorean inequality |
| Two sides and the angle between them | The third side | The law of cosines |
The converse is also what makes Pythagorean triples useful. Once you check that 82 + 152 = 172, the converse guarantees that a triangle with sides 8, 15 and 17 really has a right angle, without anyone needing to draw it.
Frequently asked questions
What is the converse of the Pythagorean theorem in simple words?
If the two shorter sides, squared and added, equal the longest side squared, the triangle has a right angle. The right angle is the corner across from the longest side. For example, 52 + 122 = 169 = 132, so a 5-12-13 triangle is right.
Is the converse of the Pythagorean theorem always true?
Yes, for triangles drawn on a flat surface. It is a proven theorem, Proposition 48 in Book I of Euclid's Elements. On a curved surface such as a sphere the rules change; see the theorem on curved surfaces.
How do I know which side is c?
Side c is always the longest of the three. If two sides tie for longest, the triangle cannot be right: with sides x, c, c, the sum x2 + c2 is always bigger than c2, so the triangle is acute.
What is the difference between the converse and the contrapositive?
The contrapositive of the theorem is "if a2 + b2 ≠ c2, the triangle is not right." It means the same as the theorem, and you use it whenever a test fails. The converse, "if a2 + b2 = c2, the triangle is right," is a separate fact, and you use it whenever a test succeeds.
Can I use the converse with decimals or square roots?
Yes, any positive lengths work. Keep values exact for as long as you can. Rounded values cause false "no" answers: 12 + 12 = 2, but 1.4142 = 1.999396, so if you round √2 to 1.414, an exactly right triangle looks slightly acute.
Is there a calculator that checks whether a triangle is right?
Yes. Enter three lengths in the right triangle checker and it sorts them, squares them and tells you whether the triangle is right, acute or obtuse, with every step shown.
Keep exploring
- 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.
- LearnThe Pythagorean Inequality: Acute, Right or Obtuse?Classify any triangle as acute, right or obtuse from its side lengths with the Pythagorean inequality: compare c² with a² + b². Proof, animation and examples.
- 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.
- TriplesPythagorean TriplesWhat Pythagorean triples are, the common ones to know (3-4-5, 5-12-13, 8-15-17…), how to generate and recognize them, and the patterns every triple shares.
- TriplesThe 3-4-5 TriangleThe 3-4-5 triangle explained: its angles (36.87° and 53.13°), area, perimeter and inradius, the 3-4-5 rule for square corners, multiples and history.
- Real-life usesThe Pythagorean Theorem in ConstructionHow builders use the Pythagorean theorem: the 3-4-5 rule for squaring corners and foundations, diagonal checks, stairs, rafters, ramps, braces and field tips.