Pythagorean Theorem Calculator
Calculate any side of a right triangle using the Pythagorean theorem. Solve for hypotenuse or either leg.
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How to use this calculator
In a right triangle, the square of the hypotenuse (c) equals the sum of squares of the two legs (a and b). Leave one field at 0 to solve for it.
- 1
Enter two known sides of the right triangle.
- 2
Set the unknown side to 0.
- 3
The calculator solves for the missing side and shows all angles and area.
- 4
The hypotenuse is always the longest side, opposite the right angle.
Frequently asked questions
What formulas does the calculator apply?
With both legs known, c = √(a² + b²). With the hypotenuse and one leg known, the other leg is √(c² − known leg²). Angle A is atan(a ÷ b) and Angle B is atan(b ÷ a), both in degrees. Area is a × b ÷ 2 and perimeter is a + b + c.
Which box do I leave at 0?
Exactly one of a, b and c must be 0, and that is the side the tool solves for. An emptied box is read as 0 as well. Two zeros, or no zero, return “Enter exactly one 0 for the unknown side”, so a full 3-4-5 entry cannot be used to check a triangle.
When does it report an error about the hypotenuse?
When you give c and one leg, c must be strictly larger than that leg. Entering b = 5 and c = 5, or b = 6 and c = 5, returns “Hypotenuse must be longer than either leg”. A leg equal to c would leave a zero-length side, which the tool does not accept.
What about negative numbers?
They are rejected with “Side lengths cannot be negative”. Previously a negative leg went through and produced a negative area and negative angles, for example −3 and 4 gave an area of −6. Lengths must be zero (for the unknown) or positive.
How are results rounded and in what units?
Sides, area and perimeter are rounded to 6 decimal places with trailing zeros dropped, and angles to 2 decimals. Values below 0.001 keep 6 significant digits, so legs of 0.0000001 and 0.0000002 return a hypotenuse of 2.23607e-7 rather than 0. There are no units: the answer is in whatever unit you typed, and the area is in that unit squared.
What are Pythagorean triples?
They are whole-number solutions such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Multiples such as 6-8-10 also qualify. Entering 5 and 12 with c at 0 returns exactly 13 and angles of 22.62° and 67.38°.
Does the tool work in three dimensions?
No, it handles one right triangle. For a box diagonal use d = √(a² + b² + c²), which you can do in two steps here: solve the floor diagonal from two edges, then use that result with the third edge as the legs of a second triangle.
Does it work for triangles without a right angle?
No. The equations assume a 90° corner and there is no angle input. For other triangles use the law of cosines, which the tool does not provide, and note that the Angle A and Angle B outputs always sum to 90°.
The Pythagorean theorem explained
Understanding a² + b² = c²
In any right triangle, the area of the square built on the hypotenuse equals the combined area of squares built on the two legs. This geometric interpretation — not just an algebraic formula — is why the theorem has such deep significance in mathematics.
The 3-4-5 trick in construction
To check if a corner is a perfect right angle: measure 3 units along one wall, 4 units along the other. If the diagonal measures exactly 5, the corner is square. This "3-4-5 check" has been used by builders for at least 4,000 years.
Non-Euclidean geometry and the theorem
The Pythagorean theorem holds exactly only on flat (Euclidean) surfaces. On a sphere (like Earth), triangles have angle sums greater than 180° and the theorem gives only an approximation. This is why GPS calculations use spherical trigonometry rather than the flat-earth Pythagorean formula.
Common Pythagorean triples
A Pythagorean triple is a set of whole numbers where a² + b² = c². Any multiple of a triple is also a triple, so 3-4-5 gives 6-8-10, 9-12-15 and so on. Enter any two sides from a row to check the third.
| a | b | c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
| 9 | 40 | 41 | 81 + 1,600 = 1,681 |
Worked example: how high does a ladder reach?
A 5 m ladder has its foot 1.5 m from a wall. The height it reaches is √(5² − 1.5²) = √22.75 = 4.7697 m. In the calculator enter b = 1.5, c = 5 and leave a at 0: it returns a = 4.769696, Angle A = 72.54° (between the ladder and the ground), Angle B = 17.46° and an area of 3.577272.
Worked example: the real size of a TV screen
Screen sizes are quoted as the diagonal. A 55-inch 16:9 TV has a width of 55 × 16 ÷ √(16² + 9²) = 47.94 in and a height of 26.96 in. Entering 47.94 and 26.96 with the hypotenuse at 0 returns 55.000775, which confirms the diagonal to within rounding of the inputs.
Common mistakes with the theorem
Using it on a triangle without a right angle: for those, use the law of cosines instead. Treating a leg as the hypotenuse: c is always the longest side, and the calculator reports an error if the hypotenuse is not longer than a leg. Mixing units, such as one side in feet and another in inches, gives a meaningless answer. Negative lengths are rejected, and a leg typed as 0 is read as the unknown, so a genuine zero-length side cannot be used.
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →