Free Pythagorean Theorem Calculator Online
Solve for any missing side of a right triangle with this free Pythagorean theorem calculator. Enter two known values, and the calculator shows the missing side, area, perimeter, and step-by-step work.
Enter two known sides to find the missing side using the Pythagorean theorem (a² + b² = c²).
Hypotenuse c
Hypotenuse c = 5
Step-by-Step
How the Pythagorean Theorem Calculator Works
The core formula is a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a, b are the two legs. Choose which side to solve for, enter the two known values, and the Pythagorean theorem calculator rearranges the equation automatically. To find c, it computes √(a² + b²). To find a leg, it computes √(c² − b²). The calculator also returns the triangle area (½ab) and the perimeter (a + b + c).
Real-World Uses of the Pythagorean Theorem
Construction workers use the Pythagorean theorem to check that corners are square. The "3-4-5 rule" is a quick field test: measure 3 feet along one wall, 4 feet along the other, and if the diagonal is exactly 5 feet, the corner is a perfect right angle. Surveyors, architects, and even hikers use the same principle to calculate distances across terrain. This Pythagorean theorem calculator makes the arithmetic instant so you can focus on the measurement.
Common Pythagorean Triples
| Side a | Side b | Hypotenuse c | Note |
|---|---|---|---|
| 3 | 4 | 5 | The classic |
| 5 | 12 | 13 | Common in construction |
| 8 | 15 | 17 | Moderate triangle |
| 7 | 24 | 25 | Elongated triangle |
| 20 | 21 | 29 | Nearly isosceles |
| 9 | 40 | 41 | Very elongated |
Beyond the Basics
The Pythagorean theorem has some surprising extensions that go well beyond finding a missing side.
Distance Formula
The distance between two points (x&sub1;, y&sub1;) and (x&sub2;, y&sub2;) is √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²). That is the Pythagorean theorem applied to coordinates.
3D Diagonal
For a box with sides a, b, c, the space diagonal is √(a² + b² + c²). This extends the Pythagorean theorem into three dimensions. The Pythagorean theorem calculator handles the 2D case; for 3D, apply the formula twice.
Triangle Type Test
If a² + b² = c², the triangle is right. If a² + b² > c², it is acute. If a² + b² < c², it is obtuse. The Pythagorean theorem provides a quick classification tool.
Frequently Asked Questions
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