Pythagorean Theorem Calculator
Solve a² + b² = c² for missing hypotenuse or leg with step-by-step radical simplification and exact decimal solutions.
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What Is a Pythagorean Theorem Calculator?
The Pythagorean theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse (c) is equal to the sum of the areas of the squares on the other two sides (legs a and b): a² + b² = c².
How to Use This Calculator
- Choose whether to solve for Hypotenuse (c) or Leg (b).
- Enter the known positive side lengths.
- View the exact simplified radical form (e.g. 5√2) and decimal approximation with full step-by-step substitution.
Pythagorean Equation
c = \sqrt{a^2 + b^2}, \quad a = \sqrt{c^2 - b^2}, \quad b = \sqrt{c^2 - a^2}To find hypotenuse c, sum the squares of the legs and take the square root. To find a leg, subtract the known leg squared from hypotenuse squared and take the root.
Worked Example
Scenario: Find hypotenuse c for legs a = 9 and b = 12
Apply theorem: c² = a² + b²
Substitute values: c² = 9² + 12² = 81 + 144 = 225
Solve for c: c = √225 = 15
Tips & Key Notes
- Hypotenuse c must always be strictly greater than either leg a or b.
- If a² + b² = c², the triangle is right-angled. If a² + b² > c², it is acute. If a² + b² < c², it is obtuse.
- Pythagorean triples (like 3-4-5 and 5-12-13) are sets of three positive integers that satisfy a² + b² = c².
Frequently Asked Questions
Can the Pythagorean theorem be used on non-right triangles?
No. The Pythagorean theorem only applies to right triangles (90° angle). For general triangles, use the Law of Cosines (c² = a² + b² - 2ab cos C).
Who proved the Pythagorean theorem?
Attributed to ancient Greek mathematician Pythagoras of Samos (c. 570–495 BC), though Babylonian and Indian mathematicians knew the relationship centuries earlier.
What is a Pythagorean triple?
A Pythagorean triple consists of three positive integers (a, b, c) such that a² + b² = c², such as (3, 4, 5) and (5, 12, 13).
How do you simplify radicals in the Pythagorean theorem?
Factor out perfect squares from under the square root. For example, √72 = √(36 × 2) = 6√2.
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