The Pythagorean theorem calculator on this page solves the single most useful relationship in geometry: for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Give it any two of the three sides and it finds the missing one, or hand it all three sides and it will tell you honestly whether they actually form a valid right triangle at all.
Arb Digital built this tool to go beyond a bare number. Alongside the missing side, you also get the triangle's perimeter, its area, both non-right angles, and a full written breakdown of the arithmetic — the same steps you'd be expected to show on a geometry assignment.
What This Pythagorean Theorem Calculator Does
Choose what you need from the dropdown menu and the calculator handles four distinct situations:
- Find the hypotenuse when you know both legs (the two shorter sides that meet at the right angle)
- Find a missing leg when you know the other leg and the hypotenuse
- Verify three known sides to check whether they truly satisfy a² + b² = c² and therefore form a genuine right triangle
- In every mode, it also reports the perimeter, the area, and the two non-right angles of the triangle, plus a step-by-step written solution
How to Use the Pythagorean Theorem Calculator
- Pick your scenario from the dropdown. Choose whether you're solving for the hypotenuse, a leg, or verifying three known side lengths.
- Enter the known side lengths into the fields that appear. The calculator only shows the inputs it actually needs for your chosen scenario.
- Click "Calculate." The missing value, perimeter, area, and both angles appear instantly.
- Read the headline result for the value you asked about, then check the result grid for the perimeter, area, and angles.
- Follow the step-by-step working below the inputs to see exactly how the theorem was applied, including the squaring, adding or subtracting, and square-rooting.
- If verifying three sides, read the sub-line carefully — it tells you exactly how the largest side squared compares to the sum of the squares of the other two, so you can see how close (or far) the triangle is from being a right triangle if it isn't one exactly.
The Pythagorean Theorem — How It's Calculated
For any right triangle with legs a and b and hypotenuse c (the side opposite the right angle), the theorem states:
a² + b² = c²
To find the hypotenuse when both legs are known, rearrange to c = √(a² + b²). To find a missing leg when the hypotenuse and the other leg are known, rearrange to a = √(c² − b²) (or b = √(c² − a²)). This relationship, named for the ancient Greek mathematician Pythagoras, is one of the most thoroughly proven results in all of mathematics — there are literally hundreds of documented independent proofs, and it forms the foundation of Euclidean geometry as described by sources like the Encyclopaedia Britannica. Because it converts side lengths of a right triangle into a simple squares-and-square-roots relationship, it underlies distance calculations in fields as varied as construction, navigation, computer graphics, and physics.
How to Verify Three Sides Form a Right Triangle
Given three side lengths, you can check whether they form a right triangle without ever drawing it. First identify the longest of the three lengths — that side would have to be the hypotenuse if a right angle exists at all. Then square all three sides and check whether the square of the longest side exactly equals the sum of the squares of the other two. If it does, the three sides form a right triangle; if it doesn't, they form some other kind of triangle (acute or obtuse) instead, or possibly no valid triangle at all if the lengths don't even satisfy the basic triangle inequality (the two shorter sides must add up to more than the longest side). This calculator performs that exact check and reports the difference between the two sides of the equation so you can see how close the triangle came to satisfying the theorem.
A handful of "Pythagorean triples" — whole-number side lengths that satisfy the theorem exactly — come up constantly in textbooks and real construction work: 3-4-5, 5-12-13, 8-15-17, and 7-24-25 are the most common. If your three sides are a multiple of one of these (like 6-8-10, which is just 3-4-5 doubled), you've got a right triangle guaranteed.
Perimeter, Area, and the Two Missing Angles
Once all three sides of a right triangle are known, the rest of the triangle's basic measurements follow directly. The perimeter is simply the sum of all three sides: a + b + c. The area of a right triangle is especially simple compared to other triangles, because the two legs are already perpendicular to each other and can be used directly as base and height: area = (a × b) / 2 — no separate height calculation needed, which is one of the reasons right triangles are so convenient to work with.
The two non-right angles can be found using basic trigonometry once all three sides are known: the angle opposite a given leg has a sine equal to that leg divided by the hypotenuse, so angle = arcsin(leg / hypotenuse). Since every triangle's interior angles sum to 180° and one angle is already fixed at 90°, the other two acute angles always add up to exactly 90° between them — a useful check to confirm your triangle's angles were computed correctly.
Real-World Uses of the Pythagorean Theorem
This isn't just a classroom exercise. Builders use it to square up walls and foundations (the classic "3-4-5 rule" for a perfect right angle), electricians and installers use it to figure the shortest cable run across a rectangular space, TV and monitor sizes are measured along the diagonal using this exact formula, and GPS and mapping software uses a version of it constantly to compute straight-line distances between coordinates. Understanding it well pays off far beyond a single math test.
The 3-4-5 Rule Builders Use on Job Sites
One of the most common practical applications of this theorem doesn't involve a calculator at all — it's the "3-4-5 rule" carpenters, framers, and landscapers use to check whether a corner is truly a 90° right angle without any specialized tools. Measure 3 units along one wall or edge from the corner, measure 4 units along the adjoining wall or edge, and then measure the straight-line distance between those two marked points. If that diagonal distance comes out to exactly 5 units (or any consistent multiple, like 6-8-10 or 9-12-15), the corner is a true right angle; if the diagonal is longer or shorter than 5 units, the corner is out of square and needs adjustment. This is exactly the same 3-4-5 Pythagorean triple this calculator uses as its default example, and it's a nice illustration of how a purely abstract-feeling algebra formula solves a very concrete, physical problem on a job site.
Beyond the Right Triangle: The Law of Cosines
The Pythagorean theorem is technically a special case of a more general rule called the Law of Cosines, which works for any triangle, not just right triangles: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. Notice that when angle C is exactly 90°, cos(90°) equals zero, and the last term drops out entirely, leaving you with the familiar a² + b² = c². That's a useful thing to know if a problem hands you a triangle that isn't guaranteed to have a right angle — the Law of Cosines will still get you the missing side, using the Pythagorean theorem's structure as its foundation.
Arb Digital builds fast, high-converting websites and content — and free tools like this one to go along with them. Browse more calculators below, or reach out if your business needs a site built with this level of care.
Try the Slope Calculator All Free ToolsCommon Mistakes to Avoid
- Mixing up which side is the hypotenuse. The hypotenuse is always the longest side and always opposite the right angle — never one of the two legs.
- Adding instead of finding the correct rearrangement. To find a missing leg, you subtract the known leg's square from the hypotenuse's square (c² − b²), you don't add them.
- Forgetting to take the square root at the end. a² + b² = c² gives you the square of the side, not the side itself — the final step is always a square root.
- Assuming any three numbers form a triangle. Three lengths only form a valid triangle at all if the two shorter sides add up to more than the longest one (the triangle inequality), separate from whether it's specifically a right triangle.
- Rounding intermediate steps too early. Keep full decimal precision until the final answer, especially when the square root doesn't come out to a whole number.
Related Free Tools From Arb Digital
Pair this calculator with our Slope Calculator for coordinate geometry, the Quadratic Equation Solver for algebraic equations, and the Scientific Calculator for any extra arithmetic. For everyday number problems, try our Fraction Calculator and Ratio Calculator, or browse everything in our free online tools hub.
Frequently Asked Questions
The hypotenuse is always the longest side of a right triangle, and it always sits directly opposite the 90° angle; the other two sides are called legs.
Yes, choose the corresponding option in the dropdown; the calculator rearranges the theorem to a = √(c² − b²) or b = √(c² − a²) as needed.
Square all three sides, then check whether the square of the longest side equals the sum of the squares of the other two; the calculator's verify mode does this automatically and shows the difference if it doesn't match exactly.
Because the two legs are already perpendicular, the area is simply half the product of the two legs: area = (a × b) / 2, with no separate height calculation needed.
Each acute angle equals the arcsine of the opposite leg divided by the hypotenuse, and the two acute angles always add up to exactly 90°.
They are sets of whole numbers, like 3-4-5 or 5-12-13, that satisfy a² + b² = c² exactly, making them common examples in textbooks and real-world right-angle construction.
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