The Pythagorean Theorem
One equation — a² + b² = c² — connects the three sides of every right triangle and unlocks a surprisingly large portion of geometry, trigonometry, and real-world measurement.
What the Theorem States
In any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides, called the legs. If the legs are labelled a and b and the hypotenuse is c, then:
This relationship is exact — not an approximation — and holds for every right triangle regardless of size or orientation.
The hypotenuse is always the longest side and always sits directly across from the 90° angle. If you cannot immediately spot the right angle, you cannot apply this theorem directly.
Finding a Missing Side: Three Cases
Case 1 — Find the hypotenuse
You know both legs and need the hypotenuse.
Example: a = 3, b = 4. Find c.
The triple 3-4-5 is the most famous Pythagorean triple — a set of whole numbers that satisfy the theorem exactly.
Case 2 — Find a missing leg
You know the hypotenuse and one leg and need the other leg. Rearrange the formula:
Example: c = 13, b = 5. Find a.
Case 3 — Non-integer answers
Most triangles will not produce whole-number answers. Leave the answer as a simplified square root unless a decimal approximation is required.
Example: a = 5, b = 7. Find c.
Common Pythagorean Triples
Memorising a few integer triples saves time on tests. All multiples of these triples also work (e.g. 6-8-10 is double 3-4-5):
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
A Visual Proof Idea
One of the most intuitive proofs uses area. Draw four identical right triangles with legs a and b and hypotenuse c. Arrange them inside a large square of side (a + b). The four triangles occupy area 4 × (½ab) = 2ab. The large square has area (a + b)² = a² + 2ab + b². The remaining inner region — a tilted square with side c — therefore has area a² + 2ab + b² − 2ab = a² + b². But that inner square's area is also c². Hence a² + b² = c².
If you are given three side lengths and asked whether they form a right triangle, plug them into a² + b² = c² (with c as the longest side). If the equation is satisfied exactly, it is a right triangle. If the left side is less than c², the angle at C is acute; if greater, it is obtuse.
Real-World Applications
The theorem is not confined to textbooks. It appears constantly in practical contexts:
- Construction and carpentry. A builder checks that a corner is square by measuring 3 feet along one wall and 4 feet along the other — if the diagonal is exactly 5 feet, the corner is 90°.
- Navigation. GPS systems use a three-dimensional extension of the theorem (adding a z² term) to calculate distances between points on Earth's surface and satellites.
- Screen sizes. A television described as "55 inches" refers to its diagonal — the hypotenuse of the rectangle formed by its width and height.
- Coordinate geometry. The distance formula between two points (x₁, y₁) and (x₂, y₂) — √[(x₂−x₁)² + (y₂−y₁)²] — is the Pythagorean theorem applied to a coordinate grid.
Extension: The 3D Version
In three dimensions, the distance from one corner of a rectangular box to the opposite corner uses all three dimensions:
This is simply the theorem applied twice: once to find the diagonal of the base, then again using that diagonal as one leg and the height as the other.
Summary
The Pythagorean theorem states a² + b² = c² for any right triangle. To find the hypotenuse, square the legs and add. To find a missing leg, subtract the known leg's square from the hypotenuse's square. Memorising a few common Pythagorean triples speeds up work considerably. Beyond the classroom, the theorem underlies construction, navigation, screen measurement, and the coordinate distance formula used throughout mathematics.