The Pythagorean Theorem: Statement, Proof and Problems

The Pythagorean theorem is one of the key results of school geometry: it ties together the three sides of any right triangle and comes up in problems on lengths, distances and areas. Below are the statement, some history, a simple proof, the converse, Pythagorean triples and worked problems.

Statement

In a right triangle, the hypotenuse squared equals the two legs squared and added together: a² + b² = c². Here a and b stand for the legs, and c for the hypotenuse, the side facing the right angle. In geometric terms, the area of a square drawn on the hypotenuse matches the combined area of the squares drawn on the two legs.

Where the Theorem Comes From

The theorem is named after Pythagoras, a Greek thinker born in about 570 BC, but the rule was known long before him. The Babylonian tablet Plimpton 322, from around 1800 BC, lists the sides of fifteen Pythagorean triples. The Indian Baudhayana Shulba Sutra and the Chinese Zhoubi Suanjing also describe the relationship, and in China the theorem is called the gougu theorem. The oldest surviving rigorous proof is in Euclid’s Elements, from around 300 BC; the book itself is the subject of Euclid’s Elements: Geometry Taught for 2,000 Years.

A Proof by Rearrangement

Draw a square whose side is a + b, then place four identical right triangles with legs a and b in its corners. A square with side c is left in the middle. Now count the big square’s area in two ways: as (a + b)², and as four triangles plus the middle square, 4 · ab/2 + c² = 2ab + c². Expanding gives a² + 2ab + b² = 2ab + c², so a² + b² = c².

The Converse

The reverse also holds: if a² + b² = c² in a triangle, then sides a and b meet at a right angle. Euclid states this as well. It lets you check whether a triangle is right-angled just by comparing the squares of its sides.

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers with a² + b² = c². The best known is 3, 4, 5; another is 5, 12, 13. Multiply every number in a triple by the same whole number and you get another triple, such as 6, 8, 10. A triple whose numbers share no common divisor greater than 1 is called primitive: 3, 4, 5 is primitive, but 6, 8, 10 is not.

Worked Problems

1. The legs are 6 and 8. The hypotenuse is √(36 + 64) = √100 = 10.
2. The hypotenuse is 13 and one leg is 5. The other leg is √(169 − 25) = √144 = 12.
3. A 5 m ladder stands 3 m from a wall. How high does its top reach? √(25 − 9) = √16 = 4 m.
4. The distance between the points (1, 2) and (4, 6) on a coordinate plane: the coordinates differ by 3 and 4, so the distance is √(9 + 16) = 5. The distance formula is the Pythagorean theorem in disguise.

Properties of triangles and other shapes are collected in Geometric Figures: Basic Concepts and Examples.

How to Approach Problems

First, find a right triangle in the problem; sometimes you have to complete one by drawing a height or a diagonal. Then decide what you know, two legs or a hypotenuse and a leg, and only then calculate. Check the answer: the hypotenuse is always longer than either leg. Our course teaches this systematically: analyzing the conditions, choosing theorems and formulas, justifying the solution and checking the answer.

Our Geometry Course

In our geometry course, students get to grips with the properties of figures and with theorems, learn to solve problems on their own and justify the solution, and prepare for tests, exams and olympiads. Lessons are open to students from age 13. A trial lesson costs $8: it is individual and lasts 45 minutes. More about the course is on the Geometry page.