Calculate the missing side of a right triangle using the Pythagorean theorem — find hypotenuse or legs instantly.
The Pythagorean Theorem Calculator finds any missing side of a right-angled triangle when two sides are known. The Pythagorean theorem is one of the most famous theorems in all of mathematics, with a history spanning over 2,500 years. It states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
This calculator works in three modes:
1. Find the hypotenuse (c) when both legs (a and b) are known
2. Find leg b when leg a and hypotenuse c are known
3. Find leg a when leg b and hypotenuse c are known
To use: enter the two known sides and leave the unknown side as 0.
Applications in Real Life
- Construction and carpentry: Ensuring walls are perfectly square (the 3-4-5 rule — if a right triangle has legs of 3 and 4, the hypotenuse must be 5, meaning the corner is exactly 90°).
- Navigation: Finding the direct distance between two points when you know the horizontal and vertical distances.
- Screen sizes: A 15-inch laptop screen has a 15-inch diagonal — if you know the width and height in inches, the Pythagorean theorem gives the diagonal.
- Staircase design: Calculating the length of the stringer (diagonal beam) given the rise and run.
- Physics: The magnitude of a vector is calculated using the Pythagorean theorem in 2D and 3D space.
Pythagorean Triples
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). These appear frequently in school problems and competitive exams because they give clean integer answers.
1. Enter two known sides: Enter the values for any two sides of the right triangle. Set the unknown side to 0.
2. Side a and Side b are the two legs (the sides that form the right angle).
3. Hypotenuse c is the longest side (opposite the right angle). Leave it as 0 to calculate it.
4. View Results: The calculator finds the missing side and also shows the area of the right triangle.
Pythagorean Theorem: a² + b² = c²
Where a and b are the two legs and c is the hypotenuse.
- Find c: c = √(a² + b²)
- Find b: b = √(c² − a²)
- Find a: a = √(c² − b²)
Area of right triangle = (a × b) / 2
For a 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5². Area = (3×4)/2 = 6 square units.
Example 1 — Find hypotenuse: a=3, b=4, c=? → c = √(9+16) = √25 = 5. This is the famous 3-4-5 Pythagorean triple.
Example 2 — Find a leg: a=?, b=12, c=13 → a = √(169−144) = √25 = 5. Another Pythagorean triple (5-12-13).
Example 3 — Ladder problem: A 10m ladder leans against a wall with its base 6m from the wall. How high does it reach? b = √(100−36) = √64 = 8m.