📏 Pythagorean Theorem Calc

The theorem

For a right triangle with legs a and b and hypotenuse c (always the longest side, opposite the right angle):

a² + b² = c²  ·  c = √(a² + b²)  ·  b = √(c² − a²)

Finding the hypotenuse

Example: legs of 3 and 4. Square them: 9 and 16. Add: 25. Take the square root: c = 5. This 3-4-5 triangle is the most famous Pythagorean triple and has been used by builders for thousands of years to lay out square corners.

Finding a leg

Example: a 10 ft ladder leans against a wall with its foot 6 ft from the wall. How high does it reach? b = √(10² − 6²) = √(100 − 36) = √64 = 8 ft.

Common Pythagorean triples

Beyond the missing side

Once all three sides are known the calculator also reports the area (½ × a × b, since the legs are perpendicular), the perimeter and the two acute angles, found with the inverse tangent: angle opposite a = tan⁻¹(a ÷ b). The two acute angles always add up to 90°.

Tips

• Use the same unit for every side – the answer comes out in that unit.
• The hypotenuse must be longer than either leg; if it isn’t, the numbers can’t form a right triangle.
• To check whether a triangle is right-angled, test whether the two shorter sides squared add up to the longest side squared.
• The distance between two points (x₁, y₁) and (x₂, y₂) is the same idea: √((x₂ − x₁)² + (y₂ − y₁)²).