Find a missing leg or hypotenuse using a² + b² = c², or check whether three side lengths form a right triangle.
Choose the unknown side, enter the other two, and calculate
Enter three side lengths to check if they form a right triangle
This Pythagorean theorem calculator finds a missing leg or hypotenuse of any right triangle using the equation a² + b² = c², and separately checks whether three given side lengths actually form a right triangle. Named after the ancient Greek mathematician Pythagoras, this is one of the oldest and most widely applied theorems in geometry — connecting algebra to the physical shape of a triangle. Whether you need a quick find-hypotenuse calculator for homework, a right-triangle calculator to verify a construction corner, or an a²+b²=c² calculator to check if a triangle is right, acute, or obtuse, this free tool shows the exact formula and substituted numbers behind every result.
The theorem describes a fixed relationship between the three sides of a right triangle: the sum of the squares of the two legs (the sides forming the 90° angle) always equals the square of the hypotenuse (the longest side, opposite the right angle). This isn't an approximation — it holds exactly for every right triangle, regardless of size, which is why it can be used both to calculate an unknown side and to test whether a triangle is right-angled at all.
This tool is built for students verifying geometry homework, builders and DIYers squaring up a wall or foundation corner, carpenters sizing a diagonal brace, surveyors and navigators computing straight-line distances, and engineers or designers who need a fast, accurate right-triangle calculation without manually rearranging the formula by hand.
Because it converts a purely geometric shape into simple algebra, the theorem underlies distance formulas, vector math, trigonometry, and countless practical measurements. A single equation — a² + b² = c² — lets you find any one side of a right triangle from the other two, or confirm that a corner is truly square, without a protractor or angle finder.
Construction crews use the 3-4-5 rule (a scaled Pythagorean triple) to square building corners. Carpenters use it to size diagonal braces. Navigators and pilots use it to compute straight-line distance from north-south and east-west offsets. TV and monitor diagonal sizes are calculated the same way from width and height. Physicists use it to resolve vector components, and the formula is the direct ancestor of the 2D and 3D distance formulas used in computer graphics and GPS mapping.
How this calculator solves for a missing side and checks for a right angle
The hypotenuse is opposite the largest angle (90°), so it must be strictly longer than either leg — a quick sanity check on any input.
Whole-number solutions like (3,4,5), (5,12,13), and (8,15,17) — plus any multiple of them, like (6,8,10) — satisfy a²+b²=c² exactly.
Comparing a²+b² to c² also classifies any triangle: equal means right, a²+b² > c² means acute, a²+b² < c² means obtuse.
From choosing a mode to reading your result
Choose "Find the Missing Side" to solve for a leg or hypotenuse, or "Check if a Triangle is Right-Angled" to test three known side lengths.
Choose whether the hypotenuse (c), leg a, or leg b is unknown from the dropdown — the input fields update automatically.
Type in the two sides you already know. Make sure all three sides use the same unit of measurement.
The calculator applies a² + b² = c², rearranged for whichever side is missing, and displays the substituted formula.
Enter all three side lengths and click "Check" to see whether they form a right, acute, or obtuse triangle.
Review the computed side length or right-triangle verdict, along with the a²+b² vs c² comparison shown beneath it.
Finding a hypotenuse, then confirming it with the right-triangle check
A carpenter has a right triangle brace with legs of 3 ft and 4 ft. What length should the diagonal (hypotenuse) piece be cut to — and does 3-4-5 actually form a right angle?
Explanation: Because 3² + 4² exactly equals 5², the 3-4-5 triangle is a genuine Pythagorean triple — this is precisely why builders measure 3 units along one wall, 4 units along a perpendicular wall, and confirm the diagonal is exactly 5 units to guarantee a true 90° corner, without needing a protractor.
What the missing-side value and the right-triangle verdict actually mean
| Comparison | Triangle Type | What It Means |
|---|---|---|
| a² + b² = c² | Right triangle | The triangle has exactly one 90° angle, opposite side c |
| a² + b² > c² | Acute triangle | All three angles are less than 90° |
| a² + b² < c² | Obtuse triangle | One angle is greater than 90°, opposite side c |
Reading the missing-side result: in Find the Missing Side mode, the number shown is the exact length (in your input unit) that completes a valid right triangle with the two values you entered.
Typical ranges: the hypotenuse must always exceed both legs individually, but must be less than their sum (a + b) — otherwise the "triangle" would be a straight line or wouldn't close at all.
How to verify manually: square your computed side, add or subtract as appropriate, and confirm the result matches a² + b² = c² using a basic calculator — this is exactly the check the calculator performs automatically in Check mode.
Where a and b and c meet the real world
Check right-triangle side and classification problems step by step.
Use the 3-4-5 rule to confirm a wall or foundation corner is exactly 90°.
Find the stringer (diagonal) length from a staircase's total rise and run.
Size a diagonal brace needed to keep a rectangular frame square.
Compute straight-line distance from north-south and east-west offsets.
The theorem underlies the coordinate distance formula used in mapping software.
Resolve a vector into components, or find a resultant displacement or force.
Calculate a TV or monitor's diagonal size from its width and height.
Compute on-screen distances and collision boundaries between objects.
Euclidean distance calculations (built on this theorem) are core to clustering algorithms.
Verify diagonal measurements on blueprints and floor plans.
Square up a picture frame, garden bed, or furniture project quickly.
Confirm square corners when laying out a field or court.
What this Pythagorean theorem calculator does well, and where it doesn't apply
How the same a²+b² vs c² comparison sorts any triangle into one of three types
| Type | Condition | Largest Angle | Example Sides |
|---|---|---|---|
| Right triangle | a² + b² = c² | Exactly 90° | 3, 4, 5 |
| Acute triangle | a² + b² > c² | Less than 90° | 4, 5, 6 |
| Obtuse triangle | a² + b² < c² | Greater than 90° | 3, 4, 6 |
Summary: This Pythagorean theorem calculator gives you an instant, free way to find a missing leg or hypotenuse using a² + b² = c², and to check whether three side lengths form a right, acute, or obtuse triangle — with the formula and steps shown. Pair it with related tools like the Distance Calculator and Triangle Calculator for a fuller picture of triangle and coordinate geometry.
Common questions about the Pythagorean theorem
Trusted educational references to go deeper on right-triangle geometry
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