The Pythagorean Theorem
The Pythagorean theorem is one of the most famous equations in mathematics: a² + b² = c². It describes a relationship between the three sides of any right triangle that has been known and used for over 4,000 years. Understanding it unlocks geometry, trigonometry, and much of physics.
In this lesson
1 What the Theorem States
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 (the legs). If the legs are a and b and the hypotenuse is c, then a² + b² = c².
The hypotenuse is always the longest side and always sits opposite the right angle (the 90° corner). The legs are the two shorter sides that form the right angle.
The most famous Pythagorean triple: 3² + 4² = 9 + 16 = 25 = 5². So a right triangle with legs 3 and 4 has hypotenuse exactly 5. Other triples: 5-12-13, 8-15-17, 7-24-25. These produce whole number answers and appear constantly in geometry problems.
2 Why It Works — The Geometric Proof
The most intuitive proof: draw a right triangle with legs a and b and hypotenuse c. Build squares on each side — a square of area a², a square of area b², and a square of area c². The theorem says the two smaller squares together equal the large square.
You can prove this by rearranging four copies of the triangle inside a large square. In one arrangement the empty space is c². In another arrangement the same empty space consists of a² and b². Since the empty space is the same either way, c² = a² + b². This visual proof requires no algebra — just area reasoning.
3 How to Use It
13812 feet4 The Converse: Identifying Right Triangles
The converse of the Pythagorean theorem is also true: if the three sides of a triangle satisfy a² + b² = c², then the triangle is a right triangle. This is how builders and surveyors check for right angles without using a protractor.
The classic construction technique: measure 3 units along one wall, 4 units along the adjacent wall. If the diagonal between those two points is exactly 5 units, the corner is a perfect right angle. This is called the 3-4-5 method and has been used in construction for thousands of years.
When checking whether three sides form a right triangle, c must be the largest value. If you write 5² + 13² = 12², you'll get 25 + 169 = 144, which is false. Always assign the longest side to c.
5 Beyond Right Triangles
The Pythagorean theorem extends naturally to three dimensions. The distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space is √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) — which is just two applications of the theorem.
In two dimensions, the distance formula between points (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²) — the direct application of the Pythagorean theorem to coordinate geometry.
The theorem connects to trigonometry through the identity sin²θ + cos²θ = 1, which is the Pythagorean theorem applied to a right triangle inscribed in the unit circle. Every trig identity ultimately traces back to this relationship.
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Practice Problems
📚 Further Reading & Resources
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