Pythagorean Theorem Calculator

What Is the Pythagorean Theorem Calculator?

The Pythagorean theorem relates the three sides of any right triangle. This tool solves for whichever side is missing, given the other two.

How to Use This Tool

Choose which side to find (a, b, or the hypotenuse c), then enter the other two. The missing side appears instantly.

The Pythagorean Theorem

Worked Example

Choosing the Missing Side

The calculator lets you pick which of the three sides to solve for: a, b or the hypotenuse c. Choosing a leg (a or b) means you must supply the hypotenuse and the other leg. The hypotenuse is always the longest side and sits opposite the right angle, so if you are unsure which measurement is c, look for the side that does not touch the right-angle corner.

Worked Examples for Each Mode

Solving for a leg. A hypothetical triangle has hypotenuse c = 13 and leg a = 5. Then b = the square root of (13² - 5²) = the square root of (169 - 25) = the square root of 144 = 12.

A non-tidy answer. With c = 10 and a = 7, b = the square root of (100 - 49) = the square root of 51, which the tool shows as 7.1414 (four decimal places).

Why Some Inputs Give No Result

When you solve for a leg, the hypotenuse must be strictly longer than the leg you enter. If it is equal or shorter, the subtraction would give zero or a negative number under the square root, and the tool shows no result. That is the calculator protecting you from an impossible triangle. If you see this, recheck which measurement is the hypotenuse.

Step-by-Step Tips

• Use the same unit for every side. The answer comes back in that unit.
• Look for common whole-number triples such as 3-4-5, 5-12-13 and 8-15-17. Multiples such as 6-8-10 work too, and recognizing them gives you a quick way to verify the tool's output.
• Check squareness on a project: measure 3 units along one wall and 4 along the other from a corner. If the diagonal between the marks is 5 units, the corner is a right angle.
• Estimate first. The hypotenuse should be less than the sum of the legs but more than either leg alone.

Common Mistakes

• Treating any long side as c. Only the side opposite the right angle counts.
• Adding the sides instead of adding their squares.
• Forgetting the final square root and reporting a squared value.
• Applying the theorem to a triangle that is not right-angled, which requires the Law of Cosines instead.

Real-World Situations

The theorem is useful when checking that a foundation or frame is square, when finding the length of a diagonal brace, when working out the shortest straight-line distance across a rectangular area, and when finding a ladder's reach on a wall. It also gives the distance between two points on a grid: the horizontal and vertical differences are the legs and the direct distance is c. For load-bearing or safety-related construction, confirm dimensions with the project plans and a qualified professional. The tool reports results rounded to four decimals, so field measurements are more likely to limit your accuracy than the calculation itself.

Frequently Asked Questions