Pythagorean Theorem Calculator

Understanding Right Triangles & The Pythagorean Formula

Named after the ancient Greek philosopher Pythagoras, the Pythagorean Theorem is a cornerstone principle of Euclidean geometry. It establishes a fixed mathematical relationship between the sides of a right triangle (a triangle containing a 90-degree angle).

The Pythagorean Equations

Depending on which side length is unknown, use the corresponding algebraic formula:

Hypotenuse (c)

c = √(a² + b²)

Leg (a)

a = √(c² - b²)

Leg (b)

b = √(c² - a²)

Manual Calculation Step-by-Step

Calculate the hypotenuse (c) for a right triangle with leg lengths a = 6 cm and b = 8 cm:

Square both leg lengths: 6² = 36 and 8² = 64.
Sum the squared values: 36 + 64 = 100.
Take the square root of the sum: √100 = 10 cm.

Common Integer Pythagorean Triples

Pythagorean triples are whole integer solutions to the equation:

(3, 4, 5): 3² + 4² = 9 + 16 = 25 = 5².
(5, 12, 13): 25 + 144 = 169 = 13².
(8, 15, 17): 64 + 225 = 289 = 17².

Frequently Asked Questions

Does the Pythagorean Theorem work for non-right triangles?

No. The Pythagorean theorem strictly applies to right-angled triangles (90°). For acute or obtuse triangles, use the Law of Cosines: c² = a² + b² - 2ab cos(C).

What is the hypotenuse of a triangle?

The hypotenuse is always the longest side of a right triangle and sits directly opposite the 90-degree right angle.

About This Calculator & Guide

About the Pythagorean Theorem— Find Hypotenuse & Triangle Legs Calculator

The Pythagorean Theorem is a fundamental relation in Euclidean geometry. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Formula & Calculation Method

The standard mathematical relationship used for this calculation is:

Hypotenuse: c = √(a² + b²); Side A: a = √(c² - b²); Side B: b = √(c² - a²)

Worked Step-by-Step Example

For a right triangle with sides a = 3 cm and b = 4 cm: c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm.

How to Use

Identify the two sides of the right triangle you already know.
Enter their values in the input fields.
The calculator solves for the third missing side length automatically.

Frequently Asked Questions

What is the Pythagorean Theorem formula?+

The Pythagorean 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.

How do you calculate the hypotenuse of a right triangle?+

To calculate the hypotenuse (c), take the square root of the sum of the squared legs: c = √(a² + b²).

What are Pythagorean Triples?+

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². Famous examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).