Pythagorean Theorem and Its Converse
The Pythagorean theorem states that in a right triangle with legs a and b and hypotenuse c, the squares of the legs add up to the square of the hypotenuse: a2 + b2 = c2. In brief: for legs 5 and 12, c2 = 25 + 144 = 169, so c = 13. The converse says that if the sides of a triangle satisfy a2 + b2 = c2, the triangle is a right triangle.
The theorem is one of the most used results in geometry. It lets you find a missing side, test whether a triangle has a right angle and compute distances. This guide explains the statement and the converse, shows a proof, works through typical problems and lists the most common Pythagorean triples.
What does the Pythagorean theorem say?
In a right triangle, the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the legs. In algebra: a2 + b2 = c2, where c is the hypotenuse. It holds only for right triangles.
From the theorem it follows that c = √(a2 + b2), a = √(c2 − b2) and b = √(c2 − a2).
How do you find a missing side?
- Identify the hypotenuse, which is opposite the right angle.
- Write a2 + b2 = c2 and substitute the known lengths.
- Solve for the unknown square.
- Take the positive square root.
Example 1: finding the hypotenuse
Legs 7 and 24: c2 = 49 + 576 = 625, so c = 25.
Example 2: finding a leg
Hypotenuse 17 and one leg 8: b2 = 289 − 64 = 225, so b = 15.
Example 3: a ladder
A 5 m ladder leans against a wall with its foot 1.4 m from the wall. The height is √(25 − 1.96) = √23.04 = 4.8 m.
Example 4: a diagonal
The diagonal of a rectangle 9 cm by 12 cm is √(81 + 144) = √225 = 15 cm.
Example 5: not a perfect square
Legs 2 and 3: c2 = 13, so c = √13 ≈ 3.61. Keep √13 for an exact answer.
What is the converse of the theorem?
The converse reverses the statement: if the sides of a triangle satisfy a2 + b2 = c2, where c is the longest side, then the triangle is right-angled, with the right angle opposite c. It also extends to other triangles:
- If a2 + b2 > c2, the triangle is acute.
- If a2 + b2 < c2, the triangle is obtuse.
Example 6: classify three triangles
Sides 6, 8, 10: 36 + 64 = 100, so right. Sides 5, 6, 7: 25 + 36 = 61 > 49, so acute. Sides 4, 5, 8: 16 + 25 = 41 < 64, so obtuse. Before applying the test, make sure that the three lengths can form a triangle: the sum of the two shorter sides must exceed the longest. For 4, 5, 8 it is 9 > 8 ✓.
Why is the theorem true?
A short proof uses areas. Place four copies of a right triangle with legs a and b and hypotenuse c inside a square of side a + b, so that the hypotenuses form a tilted square of side c in the middle. The big square has area (a + b)2. It also equals the area of the four triangles, 4 · (1/2)ab = 2ab, plus the inner square c2. So (a + b)2 = 2ab + c2. Expanding the left side: a2 + 2ab + b2 = 2ab + c2, so a2 + b2 = c2.
Many other proofs exist, using similar triangles, as, or rearrangement of areas.
Pythagorean triples table
| a | b | c | Check a² + b² = c² |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
| 12 | 35 | 37 | 144 + 1225 = 1369 |
| 9 | 40 | 41 | 81 + 1600 = 1681 |
| 28 | 45 | 53 | 784 + 2025 = 2809 |
| 11 | 60 | 61 | 121 + 3600 = 3721 |
| 16 | 63 | 65 | 256 + 3969 = 4225 |
| 33 | 56 | 65 | 1089 + 3136 = 4225 |
A triple is primitive if the three numbers have no common factor. Multiples such as 6-8-10 or 9-12-15 are right triangles too, since multiplying all sides by the same number keeps the relation true.
Where does the theorem come from?
The theorem is named after Pythagoras of Samos, but it was known before him. Babylonian tablets such as Plimpton 322, written roughly 3,800 years ago, list sets of numbers that satisfy it, and Indian and Chinese texts state it as well. It is not certain that Pythagoras himself proved it, and the earliest known proof in a deductive form appears in Euclid’s Elements. Hundreds of different proofs have been published since.
Where is the theorem used today?
The distance between two points in the plane comes straight from the theorem. Builders use the 3-4-5 rule to set right angles. GPS and navigation compute distances from coordinates, and screen sizes in inches are diagonals found with it. The theorem also defines the length of a vector.
Which mistakes are common?
Applying it to a triangle without a right angle
The equation holds only if the triangle is right-angled. For other triangles, use the law of cosines.
Forgetting to take the square root
c2 = 169 means c = 13, not 169.
Adding when you should subtract
When the hypotenuse is known, the missing leg uses a subtraction: b2 = c2 − a2.
Putting the hypotenuse in the wrong place
c is always the longest side. If your result for a leg is longer than the hypotenuse, the formula was misapplied.