The Theorem at a Glance
The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides:
a² + b² = c²
Here, c is always the hypotenuse — the longest side — while a and b are the two legs.
A Simple Visual Proof
One of the most elegant proofs requires only algebra and area:
- Draw a large square with side length (a + b).
- Inside, place four identical right triangles (sides a, b, c) in the corners.
- The inner shape left over is a square with side c, so its area is c².
- The large square's area is (a + b)² = a² + 2ab + b².
- The four triangles together have area 4 × (½ab) = 2ab.
- Subtracting: c² = a² + 2ab + b² − 2ab = a² + b². ✓
Solving for Missing Sides
Finding the Hypotenuse
If a = 3 and b = 4, then: c² = 9 + 16 = 25, so c = 5. This is the famous 3-4-5 right triangle.
Finding a Leg
If c = 13 and a = 5, then: b² = 169 − 25 = 144, so b = 12.
Pythagorean Triples
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². Knowing these saves time in exams:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 6 | 8 | 10 |
Note: Any multiple of a Pythagorean triple is also a triple. E.g., 6-8-10 is just 2 × (3-4-5).
Real-World Applications
- Construction: Builders use the 3-4-5 rule to ensure walls and foundations meet at right angles.
- Navigation: Calculating the straight-line distance between two points on a map.
- Computer graphics: Distance formulas between pixels or 3D coordinates rely on the theorem.
- Physics: Resolving force vectors into components uses right-triangle geometry.
- Surveying: Measuring land and determining property boundaries.
Extension: Distance Formula
The distance between two points (x₁, y₁) and (x₂, y₂) in the coordinate plane is a direct application of the Pythagorean theorem:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
The horizontal and vertical differences form the two legs, and the straight-line distance is the hypotenuse.
Common Mistakes to Avoid
- Misidentifying the hypotenuse: The hypotenuse is always opposite the 90° angle — it's not necessarily the side you're solving for.
- Using the theorem on non-right triangles: a² + b² = c² only holds for right triangles. For other triangles, use the Law of Cosines.
- Forgetting to square root: c² = 25 means c = 5, not c = 25.
- Rounding too early: Keep values as exact squares until the final step to avoid compounding errors.
The Pythagorean Theorem is one of mathematics' most powerful tools — simple in form, yet it underpins much of geometry, trigonometry, and beyond.