What Is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental relationship in geometry that applies to every right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This relationship has been known for over 2,500 years and is named after the ancient Greek mathematician Pythagoras, though evidence suggests that Babylonian and Indian mathematicians knew the result centuries earlier.
The theorem is remarkable because of its simplicity and its wide-ranging applications. It connects geometry to algebra, serves as the basis for the distance formula in coordinate geometry, and underpins trigonometry, physics, architecture, and navigation.
Whether you are calculating the diagonal of a TV screen, finding how far a ladder reaches up a wall, or determining the shortest path between two points, the Pythagorean theorem is the tool you reach for first.
The Formula: a² + b² = c²
a² + b² = c², where a and b are the lengths of the two legs (the sides that form the right angle) and c is the length of the hypotenuse (the longest side, opposite the right angle).It is critical to remember that c is always the hypotenuse — the side across from the 90° angle. The two legs a and b are interchangeable; it does not matter which you call a and which you call b.
Rearranging the formula
To find the hypotenuse, rearrange to c = √(a² + b²). To find a missing leg, rearrange to a = √(c² − b²).
For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5.
c = 5If you know the hypotenuse is 13 and one leg is 5, the other leg is √(169 − 25) = √144 = 12.
leg = 12Always check that you have a right triangle before applying this formula, because the theorem does not hold for non-right triangles.
A Visual Proof You Can Understand
There are over 400 known proofs of the Pythagorean theorem, but one of the most elegant is the rearrangement proof.
- Build the large square
Start with a large square with side length
(a + b). Inside it, arrange four identical right triangles (each with legs a and b, hypotenuse c) so that their hypotenuses form a smaller tilted square in the center. - Write the area two ways
The area of the large square is
(a + b)². This equals the area of the four triangles plus the area of the inner square:4 × (½ab) + c² = 2ab + c². - Expand and simplify
Expanding the left side gives
a² + 2ab + b² = 2ab + c². Subtract 2ab from both sides and you geta² + b² = c².
This proof is beautiful because it requires no advanced math — just basic algebra and the concept of area.
Another famous proof was published by U.S. President James Garfield, who used a trapezoid arrangement. The sheer number of proofs reflects how central this theorem is to mathematics.
Visualizing the proof helps you understand that the theorem is not an arbitrary rule but a logical consequence of how areas relate in right triangles.
Pythagorean Triples
a² + b² = c².The most famous triple is (3, 4, 5): since 9 + 16 = 25, this forms a valid right triangle with whole-number sides. Other common triples include (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29).
| Triple | Check | Hypotenuse² |
|---|---|---|
(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 |
Any multiple of a Pythagorean triple is also a triple. For instance, doubling (3, 4, 5) gives (6, 8, 10), and tripling gives (9, 12, 15).
Recognizing these triples can save you significant time on exams, because you can identify the missing side instantly without any calculation. If you see a right triangle with legs 6 and 8, you should immediately recognize the (3, 4, 5) pattern scaled by 2, so the hypotenuse is 10.
Teachers love to use these triples in problems, so committing the most common ones to memory is a worthwhile investment.
a = m² − n², b = 2mn, c = m² + n² for any positive integers m > n.Real-World Applications
The Pythagorean theorem appears constantly in everyday life and professional fields.
| Field | Application | How It Uses the Theorem |
|---|---|---|
| Construction | Verifying square corners | Builders use the 3-4-5 rule to verify that corners are square. If you measure 3 feet along one wall, 4 feet along the other, and the diagonal is exactly 5 feet, the corner is a perfect right angle. |
| Navigation | Distance calculation | The distance formula in coordinate geometry — d = √((x₂−x₁)² + (y₂−y₁)²) — is a direct application of the Pythagorean theorem. Pilots, sailors, and GPS systems all rely on this calculation. |
| Architecture and design | Roof pitch & ramps | Calculating roof pitch, ramp length, and diagonal bracing all require the theorem. |
| Computer graphics | Pixel distance | Finding the distance between two pixels on a screen uses the same formula. |
| Sports | Baseball diamond | In baseball, the distance from home plate to second base across the diamond (a square with 90-foot sides) is 90√2 ≈ 127.3 feet, found using the theorem. |
The Pythagorean theorem is one of those rare formulas where learning it in school directly translates to practical use throughout life.
Practice Problems with Solutions
A right triangle has legs of length 6 and 8. Find the hypotenuse.
Solution: c = √(36 + 64) = √100 = 10.
c = 10The hypotenuse of a right triangle is 15 and one leg is 9. Find the other leg.
Solution: a = √(225 − 81) = √144 = 12.
leg = 12A 10-foot ladder leans against a wall with its base 6 feet from the wall. How high up the wall does it reach?
Solution: height = √(100 − 36) = √64 = 8 feet.
height = 8 feetFind the distance between points (1, 2) and (4, 6).
Solution: d = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.
d = 5Is a triangle with sides 7, 10, and 12 a right triangle?
Solution: 7² + 10² = 49 + 100 = 149, but 12² = 144. Since 149 ≠ 144, it is not a right triangle.
Not a right triangleFor more practice or to check your work, snap a photo of any geometry problem with Solver AI for instant step-by-step help.
The Converse and Beyond
This gives you a way to test whether any triangle is a right triangle without measuring angles. Beyond the converse, the theorem generalizes in powerful ways.
Classifying triangles by sides
| Case | Condition | Triangle Type | Why |
|---|---|---|---|
| 1 | a² + b² = c² | Right | The angle opposite c is exactly 90°. |
| 2 | a² + b² < c² | Obtuse | The angle opposite c is greater than 90°. |
| 3 | a² + b² > c² | Acute | All angles less than 90°. |
Extending to three dimensions
In three dimensions, the theorem extends to d² = a² + b² + c², which gives you the diagonal of a rectangular box. The distance formula in any number of dimensions follows the same pattern.
c² = a² + b² − 2ab cos(C), which reduces to the Pythagorean theorem when angle C is 90° (since cos 90° = 0).Understanding the Pythagorean theorem deeply opens the door to these more advanced ideas and builds a strong foundation for trigonometry and analytic geometry.