Pythagorean Theorem Calculator
Calculate the hypotenuse ($c = \sqrt{a^2 + b^2}$) or missing legs of any right triangle with exact radicals, step-by-step algebraic solution, area, perimeter, angles, 3D space diagonal, and interactive proportional diagram.
Enter Both Known Legs (a & b)
Input the two perpendicular legs forming the 90° angle to calculate the hypotenuse $c$.
Proportional Triangle Visualizer
Scaled dynamicallyThe hypotenuse $c$ is opposite the 90° right angle and is always the longest side of a right triangle.
Step-by-Step Calculation
Complete Guide to the Pythagorean Theorem
The Pythagorean Theorem is one of the most foundational principles in Euclidean geometry, named after the ancient Greek philosopher and mathematician Pythagoras (c. 570 – c. 495 BC). It establishes a fundamental relationship among the three sides of any right triangle (a triangle having one interior angle of exactly 90°).
In any right triangle, the side opposite the 90° angle is called the hypotenuse ($c$), and it is strictly the longest side. The two adjacent sides that form the right angle are known as the legs ($a$ and $b$).
The Pythagorean Formulas
Find Hypotenuse (c)
Add the squares of both legs, then take the principal square root.
Find Missing Leg (a)
Subtract the square of leg $b$ from the square of hypotenuse $c$, then take the square root.
Find Missing Leg (b)
Subtract the square of leg $a$ from the square of hypotenuse $c$, then take the square root.
Primitive Pythagorean Triples Reference Chart
A Primitive Pythagorean Triple consists of three coprime positive integers $a$, $b$, and $c$ ($\gcd(a,b,c)=1$) that satisfy $a^2 + b^2 = c^2$. Any integer multiple of a primitive triple (such as 6-8-10 from 3-4-5) also forms a valid right triangle.
| Triple (a, b, c) | Leg a² | Leg b² | Sum a² + b² | Hypotenuse c² | Area (½ab) | Perimeter (a+b+c) |
|---|---|---|---|---|---|---|
| (3, 4, 5) | 9 | 16 | 25 | 25 | 6 | 12 |
| (5, 12, 13) | 25 | 144 | 169 | 169 | 30 | 30 |
| (8, 15, 17) | 64 | 225 | 289 | 289 | 60 | 40 |
| (7, 24, 25) | 49 | 576 | 625 | 625 | 84 | 56 |
| (20, 21, 29) | 400 | 441 | 841 | 841 | 210 | 70 |
| (9, 40, 41) | 81 | 1600 | 1681 | 1681 | 180 | 90 |
| (11, 60, 61) | 121 | 3600 | 3721 | 3721 | 330 | 132 |
Practical Real-World Applications
1. The 3-4-5 Rule in Construction
Carpenters, masonry workers, and foundation builders layout square walls by measuring 3 feet (or meters) along one leg, 4 feet along the other, and confirming the diagonal hypotenuse is exactly 5 feet. If the diagonal differs, the corner is out of square.
2. TV & Monitor Display Dimensions
Televisions and computer monitors are sold by their diagonal screen size (e.g. 55", 65"). Using the 16:9 widescreen aspect ratio ($a = 16k, b = 9k$), the hypotenuse is $c = \sqrt{16^2 + 9^2}k = \sqrt{337}k pprox 18.357k$, allowing calculation of exact screen width and height.
3. Ladder Safety (OSHA 4:1 Rule)
To safely reach a roof height $h$, safety standards recommend placing the base $a = h/4$ away from the wall. The required minimum ladder length is the hypotenuse $c = \sqrt{h^2 + (h/4)^2} = h\sqrt{1.0625} pprox 1.031 h$.
4. GPS Navigation & 2D/3D Euclidean Distance
The distance between two Cartesian coordinates $(x_1, y_1)$ and $(x_2, y_2)$ is directly derived from Pythagoras: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. In 3D space, this generalizes to $d = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$.
Frequently Asked Questions
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