Pythagorean Theorem Calculator
Solve for any missing side of a right triangle instantly with our free Pythagorean theorem calculator. Enter any two sides to find the third using a² + b² = c² — with step-by-step formula substitutions, a labelled triangle diagram, both angles, area, perimeter, and nearby Pythagorean triples. No signup required.
Select which side to find, enter the two known sides, and click Calculate. The calculator solves a² + b² = c² and shows angles, area, perimeter, and nearby Pythagorean triples. All calculations run instantly in your browser.
Quick Examples
Why Use Our Pythagorean Theorem Calculator?
Instant Pythagorean Theorem Calculations
Solve for any missing side of a right triangle in milliseconds. Our Pythagorean theorem calculator handles all three modes — find c, find a, or find b — with step-by-step formula substitutions shown instantly.
Secure Pythagorean Theorem Calculator Online
All Pythagorean theorem calculations run 100% locally in your browser. Your values never leave your device — use our Pythagorean theorem calculator online with complete privacy and zero data collection.
Pythagorean Theorem Calculator — No Installation
Use our Pythagorean theorem calculator directly in any browser with no downloads, plugins, or app installs required. Solve right triangle problems from any device, anywhere, with no setup needed.
Pythagorean Triples & Triangle Diagram
Every calculation shows a labelled right triangle diagram, angles, area, perimeter, and a table of nearby Pythagorean triples (integer right triangles). Primitive triples and their multiples are identified automatically.
Common Use Cases for Pythagorean Theorem Calculator
Geometry & Algebra Education
Students solving right triangle problems for homework or exams can use our Pythagorean theorem calculator to check their work and understand the step-by-step formula substitution. The triangle diagram makes the relationship between sides visual.
Construction & Carpentry
Builders use the Pythagorean theorem to check that corners are square (3-4-5 rule), calculate rafter lengths, and find diagonal measurements. Our calculator handles any side lengths with full precision.
Architecture & Engineering
Architects and structural engineers calculate diagonal bracing lengths, staircase rise-run ratios, and roof pitch using the Pythagorean theorem. Our calculator shows the hypotenuse and both angles instantly.
Programming & Game Development
Developers use the Pythagorean theorem to calculate distances between points in 2D and 3D space, collision detection radii, and vector magnitudes. Our calculator verifies distance calculations instantly.
Navigation & Surveying
Surveyors and navigators use the Pythagorean theorem to calculate straight-line distances from horizontal and vertical components. Our calculator handles any unit — metres, feet, miles, or kilometres.
Manufacturing & CNC Machining
CNC programmers and machinists use the Pythagorean theorem to calculate tool paths, diagonal cuts, and hole positions. Our Pythagorean theorem calculator provides the precision needed for manufacturing tolerances.
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean theorem states that in a 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): a² + b² = c². It is one of the most fundamental theorems in mathematics, attributed to the ancient Greek mathematician Pythagoras (c. 570–495 BCE), though it was known to Babylonian and Indian mathematicians centuries earlier. Our Pythagorean theorem calculator solves for any missing side and shows the complete triangle properties.
How Our Pythagorean Theorem Calculator Works
- 1. Select what to find: Choose whether to find the hypotenuse c (given legs a and b), leg a (given b and c), or leg b (given a and c).
- 2. Enter the two known sides: Type any positive numbers — the calculator works with integers, decimals, and irrational values like √2.
- 3. Get complete results: Click Calculate to see the missing side, both angles, area, perimeter, step-by-step solution, triangle diagram, and nearby Pythagorean triples — all processed locally in your browser.
The Three Formulas
- Find the Hypotenuse c: c = √(a² + b²) Given both legs a and b, the hypotenuse is the square root of the sum of their squares. Example: a=3, b=4 → c = √(9+16) = √25 = 5.
- Find Leg a: a = √(c² − b²) Given leg b and hypotenuse c, leg a is the square root of the difference of their squares. The hypotenuse must be greater than leg b.
- Find Leg b: b = √(c² − a²) Given leg a and hypotenuse c, leg b is the square root of the difference of their squares. The hypotenuse must be greater than leg a.
Pythagorean Triples
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². A primitive triplehas no common factor (GCD = 1). All primitive triples can be generated using Euclid's formula: a = m² − n², b = 2mn, c = m² + n² where m > n > 0, GCD(m,n) = 1, and m−n is odd.
| a | b | c | Type |
|---|---|---|---|
| 3 | 4 | 5 | Primitive |
| 5 | 12 | 13 | Primitive |
| 8 | 15 | 17 | Primitive |
| 7 | 24 | 25 | Primitive |
| 20 | 21 | 29 | Primitive |
| 9 | 40 | 41 | Primitive |
| 6 | 8 | 10 | 2×(3,4,5) |
| 5 | 12 | 13 | Primitive |
| 10 | 24 | 26 | 2×(5,12,13) |
Special Right Triangles:
- 45-45-90 triangle: Both legs are equal. If a = b = 1, then c = √2 ≈ 1.414. Sides are in ratio 1 : 1 : √2.
- 30-60-90 triangle: Sides are in ratio 1 : √3 : 2. If the short leg a = 1, then b = √3 ≈ 1.732 and c = 2.
- 3-4-5 rule: The simplest Pythagorean triple. Used in construction to verify right angles — measure 3 units along one wall, 4 along the other; the diagonal should be exactly 5.
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Frequently Asked Questions About Pythagorean Theorem Calculator
A Pythagorean theorem calculator solves for any missing side of a right triangle using a² + b² = c². Our calculator handles all three modes (find c, find a, or find b), shows step-by-step formula substitutions, a triangle diagram, angles, area, perimeter, and nearby Pythagorean triples — all processed instantly in your browser with no signup required.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the two legs (a and b): a² + b² = c². The hypotenuse is always the longest side, opposite the right angle.
To find the hypotenuse c, use the formula c = √(a² + b²). For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5. Select "Find hypotenuse c" in our Pythagorean theorem calculator, enter a and b, and click Calculate.
To find leg a given b and c: a = √(c² − b²). To find leg b given a and c: b = √(c² − a²). The hypotenuse must always be greater than either leg. Select the appropriate mode in our calculator and enter the two known values.
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous is (3, 4, 5). A primitive triple has no common factor — (3,4,5) is primitive, but (6,8,10) is not (it is 2×(3,4,5)). Our calculator shows nearby Pythagorean triples for every calculation.
The 3-4-5 rule is a practical application of the Pythagorean theorem used to check right angles in construction. Measure 3 units along one wall from the corner, 4 units along the adjacent wall, and the diagonal between those two points should be exactly 5 units. Any multiple works: 6-8-10, 9-12-15, etc.
Yes, completely. All Pythagorean theorem calculations run 100% locally in your browser using JavaScript. Your values are never sent to any server, stored, or logged.
Yes. Our Pythagorean theorem calculator is 100% free with no signup, no account, no premium tier, and no usage limits. Calculate right triangle sides as many times as you need — completely free, forever.
The two most common special right triangles are the 45-45-90 triangle (sides in ratio 1:1:√2) and the 30-60-90 triangle (sides in ratio 1:√3:2). These appear frequently in geometry, trigonometry, and engineering. Enter a=1, b=1 for the 45-45-90 triangle, or a=1, b=1.732 for the 30-60-90 triangle in our calculator.