Pythagorean Theorem Calculator
Solve for the hypotenuse or legs of a right triangle. Get simplified radical values and step-by-step arithmetic proofs.
Solved Hypotenuse c
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Understanding the Pythagorean Theorem (a² + b² = c²)
Named after the ancient Greek mathematician Pythagoras, this fundamental geometric theorem states that in any right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two legs (a and b).
a² + b² = c²
The hypotenuse is always the longest side and lies directly opposite the 90-degree right angle.
Practical Applications
- Construction: Carpentry framing professionals use the 3-4-5 rule to build perfectly square corners.
- Cartesian Coordinate Systems: The distance between coordinates (x1, y1) and (x2, y2) is solved using the Pythagorean-derived Euclidean formula.
- Screen Sizes: Television and computer monitor sizes are declared by the diagonal hypotenuse length.
Frequently Asked Questions About Pythagorean Theorem
What is the Pythagorean Theorem formula?
The Pythagorean Theorem is written as a^2 + b^2 = c^2, where 'a' and 'b' represent the two legs that meet at a 90-degree right angle, and 'c' represents the hypotenuse, which is the longest side opposite the right angle.
Can a leg of a right triangle be longer than the hypotenuse?
No. The hypotenuse is always the longest side of a right triangle because it lies opposite the largest angle (90 degrees). If you input a leg value larger than or equal to the hypotenuse, the calculator will flag it as an invalid geometry.
How do you simplify radicals in the theorem results?
Radicals are simplified by factoring out the largest perfect square factor from the number inside the square root. For example, sqrt(20) factors into sqrt(4 × 5), which simplifies to 2 × sqrt(5).
Can this find any missing side of a triangle?
Yes. Solve for hypotenuse or any missing leg with simplified radical forms.