The Pythagorean Theorem
a² + b² = c² is probably the most famous equation in mathematics. It's been known for over 4,000 years, has hundreds of proofs, and appears constantly in geometry, trigonometry, physics, and computer graphics. Here's what it actually says and why it works.
In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The hypotenuse is the longest side, opposite the right angle. If the shorter sides are a and b and the hypotenuse is c, then a² + b² = c². Classic example: a=3, b=4, c=5. Check: 9 + 16 = 25.
What the theorem actually says
A right triangle has three sides. Two of them , the legs , form the right angle. The third , the hypotenuse , is always the longest and sits opposite the right angle.
The theorem says: take each leg, square it, add those squares together, and you get the square of the hypotenuse. Always. For any right triangle, no matter the size.
The 3-4-5 triangle is the classic example because everything works out to whole numbers: 3² + 4² = 9 + 16 = 25 = 5². These whole-number combinations are called Pythagorean triples. Others include 5-12-13 (25+144=169), 8-15-17, and 7-24-25. Construction workers use the 3-4-5 method to check for right angles , if the measurements work out, the corner is square.
Why it works , the geometric proof
There are over 370 known proofs of the Pythagorean theorem, compiled in Elisha Scott Loomis's 1927 book The Pythagorean Proposition. Here's the most visually satisfying one, which requires no algebra.
Build a square on each side of the right triangle. The theorem says the area of the square on the hypotenuse equals the combined area of the squares on the two legs. You can actually verify this by cutting and rearranging the smaller squares , their pieces fit exactly into the larger square.
A simpler algebraic version: take four copies of the triangle and arrange them inside a large square in two different ways. In one arrangement, the empty space is c². In the other arrangement, the same empty space consists of two pieces: a² and b². Same space, same area, so c² = a² + b².
The proof works because the area of a square is just the side length squared , so "the square on the hypotenuse" literally means a square with side length c, which has area c².
Using the theorem to solve problems
The three typical problems: find the hypotenuse given both legs, find a leg given the hypotenuse and the other leg, or check whether a triangle is a right triangle.
c must be the hypotenuse , the longest side, opposite the right angle. If you accidentally assign c to a leg, the formula gives a wrong answer. Always identify which angle is the right angle first, then label the side opposite it as c.
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Testing whether a triangle has a right angle
The converse of the theorem is also true: if a² + b² = c², then the triangle has a right angle opposite the side of length c. This lets you check for right angles using only measurements.
Builders use this constantly. Lay out 3 feet along one wall, 4 feet along the adjacent wall, and check if the diagonal between those two points is exactly 5 feet. If it is, the corner is a perfect 90 degrees. This technique has been used in construction for thousands of years.
To check whether three side lengths form a right triangle, square all three, then see if the two smaller squares add up to the largest. Sides 5, 6, 9: 25 + 36 = 61, not 81. Not a right triangle. Sides 9, 40, 41: 81 + 1600 = 1681 = 41². Right triangle.
Where this shows up beyond basic geometry
The distance formula in coordinate geometry is just the Pythagorean theorem: the distance between (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²). The horizontal and vertical differences are the legs, the distance is the hypotenuse.
In 3D, the distance between two points extends naturally: √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Two applications of the theorem stacked together.
Trigonometry is built on the Pythagorean theorem. The identity sin²θ + cos²θ = 1 is the theorem applied to a right triangle inscribed in the unit circle. Every trigonometric identity traces back to this relationship.
In physics, whenever you break a vector into components , velocity, force, acceleration , you're using the Pythagorean theorem to find the magnitude from the components. A car going 30 mph east and 40 mph north has a speed of √(900+1600) = √2500 = 50 mph. Same 3-4-5 relationship.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.