The Pythagorean theorem in a nutshell

The Pythagorean theorem describes the relationship between the sides of a right triangle. If \(a\) and \(b\) are the legs and \(c\) is the hypotenuse (the side opposite the right angle), then

This means that the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is one of the most widely used results in geometry, physics, engineering, navigation and everyday construction.

Solving for different sides

Depending on which side is unknown, you can rearrange the formula:

• To find the hypotenuse \(c\): \(c = \sqrt{a^2 + b^2}\).
• To find a leg \(a\): \(a = \sqrt{c^2 - b^2}\).
• To find the other leg \(b\): \(b = \sqrt{c^2 - a^2}\).

In all cases, \(c\) must be the longest side; otherwise, the triangle cannot be right-angled.

Worked example (3–4–5 triangle)

Suppose you have a right triangle with legs \(a = 3\) and \(b = 4\). The hypotenuse is:

Area and perimeter of a right triangle

Once you know all three sides, you can immediately compute:

• Area: \(\dfrac{1}{2} a b\).
• Perimeter: \(P = a + b + c\).

The calculator shows these values automatically, using the same units as your inputs.

When can you use the Pythagorean theorem?

The Pythagorean theorem is only valid for right triangles (one angle is exactly \(90^\circ\)). Common uses include:

• Finding a ladder length safely reaching a given height.
• Checking whether a corner is square in construction.
• Computing the diagonal of a rectangle or screen.
• Distance between two points on a coordinate grid.

For non-right triangles, you generally need the law of cosines, which reduces to the Pythagorean theorem when the angle is \(90^\circ\).

Connection with the distance formula

In coordinate geometry, the distance between points \((x_1, y_1)\) and \((x_2, y_2)\) is

This is just the Pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse.

Pythagorean theorem – FAQ

What is the Pythagorean theorem?

It is a statement about the lengths of the sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the legs, \(a^2 + b^2 = c^2\).

How do I know which side is the hypotenuse?

The hypotenuse is always the side opposite the right angle and is always the longest side of a right triangle. In the calculator and formulas on this page, the hypotenuse is always called \(c\).

What units does the calculator use?

You can use any unit of length (meters, centimeters, inches, feet, etc.), as long as all sides use the same unit. The calculator carries the unit through to the results for area and perimeter.

Can the Pythagorean theorem be used in 3D?

Yes, but you apply it step by step. For example, to find the space diagonal of a box with sides \(x\), \(y\), and \(z\), first use the theorem in the base to get the diagonal \(d = \sqrt{x^2 + y^2}\), then use it again with \(d\) and \(z\) to get the full diagonal \(\sqrt{x^2 + y^2 + z^2}\).

What if my numbers don’t satisfy a² + b² = c² exactly?

In real measurements you usually have rounding and small errors. The calculator reports how closely \(a^2 + b^2\) matches \(c^2\) and flags impossible combinations (for example, if a leg is longer than the hypotenuse).