Pythagorean Theorem Calculator

This Pythagorean theorem calculator finds the missing side of a right triangle when the lengths of the other two sides are known. It applies the Pythagorean theorem formula, a² + b² = c², to solve for either leg or the hypotenuse depending on which values are entered. This calculation is used in construction to confirm right-angle measurements on a job site, in navigation to determine straight-line distance between two points, and in geometry coursework where students need to verify or derive triangle dimensions without working through the algebra by hand.

A right triangle is a triangle that contains one 90-degree angle, also called a right angle. The two sides that form this angle are known as the legs, typically labeled a and b. The side opposite the right angle, always the longest of the three sides, is the hypotenuse, labeled c.

The Pythagorean theorem describes the fixed relationship between these three sides. It states that the square of the hypotenuse equals the sum of the squares of the two legs:

This holds true for every right triangle, regardless of its size or proportions. If the lengths of any two sides are known, the third can be calculated directly from this equation.

The theorem is named after Pythagoras, a Greek mathematician who lived in the sixth century BC. Evidence suggests the relationship was already in use by Babylonian and Indian mathematicians before his time, but the theorem carries his name due to its association with the Pythagorean school. It remains one of the most widely applied principles in geometry and trigonometry.

To find a missing side, enter values for any two of the three sides. The calculator will solve for the third automatically.

Only two known values are required to solve the triangle. Entering just one side length will not produce a result, since a single measurement does not provide enough information to determine the other two under the Pythagorean theorem. If you know an angle instead of a second side length, you will need a right triangle calculator that accepts angle inputs rather than this one.

The calculator requires two of the three side lengths to solve a right triangle. Which formula it applies depends on which side is missing.

Solving for the hypotenuse. When both legs are known, the hypotenuse formula is used:

The two leg values are squared, added together, and the square root of that sum gives the length of the hypotenuse.

Solving for a leg. When the hypotenuse and one leg are known, the equation is rearranged to isolate the missing leg:

The known leg is squared and subtracted from the squared hypotenuse, and the square root of the result gives the length of the missing leg.

In each case, the result represents the length of the unknown side, expressed in whatever unit was selected for the input values. Changing the unit before entering values changes the unit of the output as well.

Some versions of this calculator also return two additional values once all three sides are known:

These are not part of the Pythagorean theorem itself but are commonly included since they can be calculated directly once all three side lengths are established.

Consider a right triangle with leg a = 3 and leg b = 4. To find the hypotenuse, substitute these values into the formula:

This is one of the most recognized right triangle examples, often called a 3-4-5 triangle because all three sides are whole numbers.

The same method applies when solving for a leg instead of the hypotenuse. If the hypotenuse and one leg are known, the values are substituted into the rearranged formula, squared, and subtracted before taking the square root, following the same step by step process shown above. Only the position of the known and unknown values changes.

A Pythagorean triple is a set of three positive whole numbers that satisfy the equation a² + b² = c². When all three side lengths of a right triangle happen to be integers, the triangle is described as a Pythagorean triple.

The most common examples include:

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17

Each of these sets can be scaled up by multiplying all three numbers by the same factor. For example, doubling the 3-4-5 triple gives 6, 8, 10, which is also a valid right triangle.

Pythagorean triples are useful for checking a calculation quickly without running the full formula. If a set of three side lengths matches a known triple, or a multiple of one, the triangle can be confirmed as a right triangle by recognition alone. This is a practical shortcut in fields like construction, where a 3-4-5 ratio is commonly used to square corners without measuring angles directly.

The Pythagorean theorem is applied wherever a right angle needs to be confirmed or a distance needs to be calculated without a direct measurement.

Construction. Builders use the theorem to check that corners and frames are square, and to calculate diagonal bracing lengths where a direct measurement is not practical. A diagonal length calculator based on this formula is a standard tool on job sites for confirming that a rectangular structure has true right angles.

Navigation. When movement occurs along two perpendicular paths, such as traveling a set distance east and then north, the theorem gives the shortest straight-line distance between the starting and ending points. This type of navigation distance calculation is a direct application of the distance formula, which is itself derived from the Pythagorean theorem.

Screen and display sizing.Screen diagonal calculation relies on the same principle. A display's advertised size refers to the diagonal measurement, which can be calculated from the screen's width and height using a² + b² = c².

Geometry and trigonometry coursework. The theorem is a foundation for right triangle trigonometry, where it supports concepts such as sine, cosine, and tangent ratios. Students encounter it early in geometry and continue to rely on it throughout more advanced coursework.