What is the Pythagorean Theorem formula?

For any right triangle with perpendicular legs a and b and hypotenuse c: a^2 + b^2 = c^2. Therefore, c = sqrt(a^2 + b^2).

How do you calculate a missing leg (a or b)?

To find leg a given hypotenuse c and leg b: a = sqrt(c^2 - b^2). The hypotenuse c must always be strictly greater than either leg.

What are Pythagorean Triples?

Pythagorean Triples are sets of three positive integers that perfectly satisfy a^2 + b^2 = c^2, such as (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Is this right triangle calculator free?

Yes, 100% free with area and perimeter calculations included.

Pythagorean Theorem Calculator - Solve Hypotenuse & Legs

What is the Pythagorean Theorem?

The Pythagorean Theorem is one of the most celebrated fundamental relations in Euclidean geometry. It 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 legs ($a$ and $b$):
$$a^2 + b^2 = c^2$$ It is universally applied across construction carpentry (the 3-4-5 squaring rule), navigation distance formulas, surveying, physics vectors, and computer graphics.

Key Capabilities

Hypotenuse & Leg Dual Solver: Calculate missing hypotenuse ($c = sqrt{a^2 + b^2}$) or unknown leg ($a = sqrt{c^2 - b^2}$).
Area & Perimeter Inclusions: Automatically derives total perimeter ($a+b+c$) and surface area ($rac{1}{2}ab$).
Visual Diagram: SVG right triangle illustrating exact proportional side alignments.
Step-by-Step Proof: Displays squares and square root derivations clearly.

How to Use the Pythagorean Calculator

Select whether you want to solve for the Hypotenuse (c) or a Leg (a/b).
Enter the two known side lengths.
Read the missing side length, triangle area, and perimeter.

Frequently Asked Questions

Can the hypotenuse ever be shorter than a leg?

No! The hypotenuse is opposite the 90-degree right angle (the largest angle in the triangle), so it must always be strictly longer than either leg ($c > a$ and $c > b$).

What is the 3-4-5 rule in construction?

Builders measure 3 feet along one wall and 4 feet along the adjacent wall. If the diagonal hypotenuse is exactly 5 feet, the corner is guaranteed to be a perfect 90-degree square.

Does the theorem work for non-right triangles?

No. For oblique triangles without a 90-degree angle, you must use the Law of Cosines ($c^2 = a^2 + b^2 - 2abcos C$).

Is this tool free?

Yes, it is 100% free with unlimited calculations.

Pythagorean Theorem (a² + b² = c²) Calculator

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Step-by-Step Pythagorean Solution: