Pythagorean Theorem Calculator
Find missing leg or hypotenuse ($a^2 + b^2 = c^2$), interior angles, triangle area, and perimeter with visual step-by-step solutions.
Right Triangle Solver
Enter any 2 values to calculate the missing side, angles, area, and perimeter.
How the Pythagorean Theorem Works
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 ($a$ and $b$):
- Solving Hypotenuse (c): $c = \sqrt{a^2 + b^2}$
- Solving Leg (a): $a = \sqrt{c^2 - b^2}$
- Solving Leg (b): $b = \sqrt{c^2 - a^2}$
- Area (A): $\text{Area} = \frac{1}{2} \times a \times b$
- Perimeter (P): $P = a + b + c$
- Angles ($\alpha, \beta$): $\alpha = \arcsin(a/c)$, $\beta = \arcsin(b/c)$
Frequently Asked Questions (FAQs)
It is used to calculate the length of an unknown side of a right triangle when two other side lengths are known. It is widely applied in geometry, construction, navigation, surveying, and computer graphics.
Pythagorean triples are sets of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Common examples include $(3, 4, 5)$, $(5, 12, 13)$, and $(8, 15, 17)$.
No, the Pythagorean theorem only applies strictly to right triangles (triangles with one 90-degree angle). For non-right triangles, you must use the Law of Cosines or Law of Sines.
Once you know side lengths $a$, $b$, and $c$, you can use inverse trigonometric functions: angle $\alpha = \arcsin(a/c)$ and angle $\beta = \arcsin(b/c)$, with the third angle always being $90^\circ$.
In any triangle, the longest side is opposite the largest angle. In a right triangle, the right angle ($90^\circ$) is the largest angle, making hypotenuse $c$ always strictly greater than legs $a$ or $b$.