Pythagorean Theorem Calculator
Solve for any side ($a$, $b$, or $c$) of a right triangle using the famous formula: $$a^2 + b^2 = c^2$$
Enter Two Known Side Lengths
Calculation Results
Input any two sides of the right triangle above to find the missing side length and the area.
Important Info: Understanding the Pythagorean Theorem
The Pythagorean Theorem is a fundamental relationship in Euclidean geometry among the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).
In mathematical terms, for a right triangle with legs $a$ and $b$ and hypotenuse $c$, the formula is always:
A right triangle is defined as a triangle with one angle exactly equal to $90^\circ$ (a right angle). The sides forming the right angle are called the legs ($a$ and $b$), and the side opposite it is the hypotenuse ($c$). [Image of a right triangle with sides a, b, and c labeled]
Why Use This Calculator?
This theorem is not just abstract math; it has countless practical applications:
- **Construction and Carpentry:** Determining the necessary length of diagonal braces for rectangular frames (like squaring walls).
- **Navigation and Surveying:** Calculating the shortest distance between two points that are not directly accessible, or determining the distance of a ship from shore.
- **Computer Graphics and Games:** Calculating distances between objects in a 2D or 3D coordinate system.
- **Roofing and Architecture:** Finding the length of a roof's diagonal span (rafter length) based on the rise and run.
How to Use This Calculator
- Identify Known Sides: You must know the length of any two sides of the right triangle.
- Input Values: Enter the lengths into the corresponding fields ($a$, $b$, or $c$). Leave the unknown side field blank.
-
Calculate: Click the "Calculate Missing Side" button. The tool will automatically detect which side is missing and apply the correct formula derived from $a^2 + b^2 = c^2$:
- To find the hypotenuse ($c$): $\sqrt{a^2 + b^2}$
- To find a leg ($a$ or $b$): $\sqrt{c^2 - b^2}$ or $\sqrt{c^2 - a^2}$
- Review Results: The result card will show the calculated missing side length, the area of the triangle, and the perimeter.
Frequently Asked Questions (FAQ)
What is a Pythagorean Triple?
A Pythagorean triple is a set of three positive integers $a$, $b$, and $c$, such that $a^2 + b^2 = c^2$. The most famous example is $(3, 4, 5)$, because $3^2 + 4^2 = 9 + 16 = 25$, and $5^2 = 25$. This calculator handles all numbers, not just integers.
Can I use this for non-right triangles?
No. The Pythagorean theorem is strictly valid only for right-angled triangles. For non-right triangles, you would use more advanced concepts like the Law of Sines or the Law of Cosines.
Why isn't there a currency converter option?
This is a purely geometric and mathematical calculation dealing with lengths and dimensions, which are unit-agnostic (e.g., feet, meters, inches). Currency conversions are not relevant here, but you can use the resulting length in any unit you prefer!