What is Pythagorean Theorem Calculator?
The Pythagorean theorem is one of the most fundamental and widely used results in all of mathematics. It states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c².
This calculator lets you solve a right triangle in three ways: find the hypotenuse when you know both legs, find one leg when you know the other leg and the hypotenuse, or verify whether three given sides form a valid right triangle. Every result includes the area, perimeter, and a complete step-by-step derivation so you can follow the algebra.
Steps:
- Choose what you want to find: the hypotenuse or one of the legs.
- Enter the two known side lengths.
- The calculator applies the Pythagorean theorem and shows the full algebraic derivation.
- Review the area and perimeter of the right triangle.
Formula
Pythagorean Theorem: a² + b² = c²
Finding the hypotenuse: c = √(a² + b²)
Finding leg a: a = √(c² - b²)
Finding leg b: b = √(c² - a²)
Area: A = ½ab
Perimeter: P = a + b + c
Use Cases
- Solving geometry homework and exam problems
- Construction — calculating diagonal distances
- Navigation — finding straight-line distances between two points
- Architecture — verifying right angles in building layouts
- Physics — resolving vector components
Key Benefits
- Three solving modes: find hypotenuse, find leg a, or find leg b
- Animated SVG right triangle visualization with labeled sides
- Full step-by-step algebraic derivation using KaTeX-rendered formulas
- Instant area and perimeter calculation
- Copy, share, and export results with one click
Pro Tips
- Use Pythagorean triples (3-4-5, 5-12-13, 8-15-17) to quickly check your work
- The hypotenuse is always the longest side of a right triangle
- If a² + b² > c², the triangle is acute; if a² + b² < c², it is obtuse
- For coordinate geometry, the distance between two points is just the hypotenuse of a right triangle formed by the horizontal and vertical differences
Common Mistakes to Avoid
- Applying the theorem to non-right triangles — it only works for 90° angles
- Confusing which side is the hypotenuse — it is always the longest side, opposite the right angle
- Subtracting when you should add (or vice versa) — remember: c² = a² + b² to find c, but a² = c² − b² to find a leg
- Forgetting to take the square root after adding or subtracting the squares
Key Terms Explained
- Hypotenuse: The longest side of a right triangle, opposite the right angle
- Leg: Either of the two shorter sides of a right triangle that form the right angle
- Pythagorean Triple: Three positive integers a, b, c that satisfy a² + b² = c²
- Right Triangle: A triangle with exactly one 90° angle
- Square Root: The inverse operation of squaring a number
Example
For a right triangle with legs a = 3 and b = 4: c = √(3² + 4²) = √(9 + 16) = √25 = 5. Area = ½ × 3 × 4 = 6 square units. Perimeter = 3 + 4 + 5 = 12 units.