The Pythagorean Theorem: Proof, Formula, and Real-World Uses
The Pythagorean theorem is one of the most famous results in all of mathematics — simple enough to state in one line, profound enough to have over 370 known proofs. Here's everything you need to know, from the formula to practical applications.
The Formula
For any right triangle (a triangle with one 90° angle):
a^2 + b^2 = c^2
Where a and b are the two legs (the sides that form the right angle) and c is the hypotenuse (the side opposite the right angle — always the longest side).
Finding Each Side
Finding the hypotenuse (c):
c = √(a^2 + b^2)
Finding a leg (a):
a = √(c^2 - b^2)
Finding the other leg (b):
b = √(c^2 - a^2)
Worked Examples
Example 1: A right triangle has legs of 3 cm and 4 cm. Find the hypotenuse.
- c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
Example 2: A ladder 10 metres long leans against a wall, with its base 4 metres from the wall. How high does it reach?
- a = √(10² - 4²) = √(100 - 16) = √84 ≈ 9.17 metres
Pythagorean Triples
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². These come up often in problems and are worth memorising:
| a | b | c | |---|---|---| | 3 | 4 | 5 | | 5 | 12 | 13 | | 8 | 15 | 17 | | 7 | 24 | 25 | | 20 | 21 | 29 |
Any multiple of a triple is also a triple: (6, 8, 10), (9, 12, 15), (15, 20, 25) all work.
A Simple Proof
The most elegant proof uses areas. Draw a large square with side (a + b). Inside it, arrange four copies of the right triangle with legs a and b.
The four triangles take up area 4 × (½ab) = 2ab. The remaining space in the large square must be c² (the square on the hypotenuse).
The large square has area (a + b)² = a² + 2ab + b².
So: a² + 2ab + b² − 2ab = c²
Therefore: a² + b² = c²
Real-World Applications
Construction and Carpentry
The "3-4-5 rule" is used every day on construction sites to check that corners are perfectly square. Measure 3 units along one wall, 4 units along the adjacent wall, then check that the diagonal measures exactly 5 units. If it does, the angle is exactly 90°.
Navigation
Before GPS, navigators used the theorem constantly. If you travel 30 km east then 40 km north, your straight-line distance from the start is √(30² + 40²) = √2500 = 50 km.
Modern GPS systems use a 3D extension of the theorem to calculate distances between coordinates.
Screen Sizes
A "65-inch TV" has a diagonal screen measurement of 65 inches. If you know the aspect ratio (16:9), you can find the exact width and height using the theorem. For a 65" 16:9 screen: width ≈ 56.7", height ≈ 31.9".
Engineering and Physics
The theorem is foundational in structural engineering (calculating load-bearing diagonals), computer graphics (rendering 3D scenes), and physics (calculating resultant vectors — the combined effect of two forces at right angles).
The 3D Extension: Distance Formula
The Pythagorean theorem extends naturally to three dimensions:
d = √((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2)
This is used in computer graphics, physics simulations, GPS calculations, and any system working with 3D coordinates.
Calculate the Pythagorean Theorem Now
Use our free calculator to find any side of a right triangle given the other two. Enter any two sides and get the third instantly, along with step-by-step working.