Pythagorean Theorem Guide: a² + b² = c² Explained勾股定理指南：a² + b² = c² 详解
The Pythagorean theorem is the oldest mathematical formula still in daily use — over 2,500 years old, and yet every carpenter, every navigator, every video game programmer still relies on it. The statement is short: in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. The implications are vast. This guide explains what the theorem really says, when it applies (and when it doesn't), and how to use it confidently.
The Theorem in Plain Language
For any right triangle (a triangle with one 90° angle), the three sides obey:
- a and b — the two shorter sides (legs)
- c — the longest side, opposite the right angle (hypotenuse)
Works in any unit system, as long as both legs use the same unit.
The converse is also true: if a triangle's sides satisfy a² + b² = c², it's a right triangle. Both directions are useful.
The Classic 3-4-5 Triangle
The most famous example: a triangle with legs 3 and 4 has hypotenuse 5:
3² + 4² = 9 + 16 = 25 = 5² ✓
The 3-4-5 triangle (and its multiples: 6-8-10, 9-12-15, 5-12-13, 8-15-17) is so well-known that carpenters carry a "speed square" to lay out perfect right angles using these proportions. Tape measures often have 3-4-5 markings printed on them.
Solving for Any Unknown Side
You can rearrange the theorem to solve for any side if you know the other two:
Worked example: you're building a ramp up to a 5-foot-high deck. The ramp must be at least 12 feet long for safety. How far from the deck should it start?
a = √(12² − 5²) = √(144 − 25) = √119 ≈ 10.9 feet
So the ramp's base sits 10.9 feet from the deck, the ramp rises 5 feet, and the ramp itself is 12 feet long — a right triangle confirmed by the Pythagorean theorem.
When the Theorem Works (and Doesn't)
The Pythagorean theorem applies only to right triangles. For other triangles, the relationship is governed by the Law of Cosines, which generalizes the Pythagorean theorem:
For a right angle, cos(90°) = 0, so the term drops out and we get the Pythagorean theorem back. For other angles, the cosine term adjusts the relationship.
For an equilateral triangle with all 60° angles, the law of cosines gives c² = a² + b² − ab, which is very different from a² + b² = c².
Distance Between Two Points
The most useful real-world application: the distance between any two points in 2D space. If you have point (x₁, y₁) and point (x₂, y₂), the straight-line distance is:
This is just the Pythagorean theorem: the differences in x and y are the legs, and the distance is the hypotenuse.
Worked example: what's the distance between (1, 2) and (4, 6)?
d = √[(4 − 1)² + (6 − 2)²] = √[9 + 16] = √25 = 5
GPS, video games, computer graphics, sports analytics, robotics — every distance calculation is the Pythagorean theorem in disguise.
The Limits — When It Breaks Down
Three cases where the theorem doesn't apply directly:
- Non-right triangles. As above — use the Law of Cosines instead.
- Curved surfaces. On a sphere, "straight lines" become great circles, and the geometry is non-Euclidean. Airplane flight paths use spherical trigonometry, not flat Pythagoras.
- Higher dimensions. The theorem generalizes: in 3D, d = √(Δx² + Δy² + Δz²); in n-D, d = √(Σ Δxᵢ²). The principle scales.
The "triangle inequality" says any side must be shorter than the sum of the other two. The Pythagorean theorem gives the special case where one side is exactly the square root of the sum of the squares of the other two — i.e., the most "efficient" triangle for given leg lengths.
Pythagorean Triples — Integer Solutions
Some right triangles have all-integer sides — they're called Pythagorean triples:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
| 9 | 40 | 41 |
| 12 | 35 | 37 |
| 11 | 60 | 61 |
There's no upper limit; an infinite family exists, generated by the formula m² − n², 2mn, m² + n² for any positive integers m > n.
Try It
Use the Pythagorean Theorem Calculator to:
- Solve for any unknown side given the other two.
- Compute the distance between two points in 2D or 3D.
- Verify whether a triangle is a right triangle by checking its sides.
The calculator works in any unit and produces exact answers when possible (for Pythagorean triples) and decimal approximations otherwise.