Pythagorean Theorem Calculator
a² + b² = c² with every step: exact surds, the converse, box diagonals and triples.
Result
This is the last result worked out. Fix the input above to update it.
Step by step
Triples
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As the hypotenuse
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About the Pythagorean Theorem Calculator
Type two sides of a right triangle and get the third with the Pythagorean theorem, a² + b² = c², worked step by step. The answer comes as a simplified surd as well as a decimal — legs of 4 and 6 give a hypotenuse of √52 = 2√13 ≈ 7.2111 — and it stays exact when you type fractions or roots such as 1/3 or √2. You also get the triangle’s angles, area, perimeter and the height to the hypotenuse, with a drawing that can show the squares on the three sides.
Three more modes use the same idea. Check a triangle applies the converse: for any three lengths it compares the square of the longest side with the sum of the other two squares, and tells you whether the triangle is right-angled, acute or obtuse. Box diagonal finds the diagonals of a box or room in two Pythagoras steps. Pythagorean triples lists every whole-number triple up to a size, builds one from Euclid’s formula, or finds all the triples that contain a number you choose.
How to use it
- Choose “Find a side”, type the two sides you know, and leave the one you want empty — the hypotenuse c is the side opposite the right angle.
- Read the missing side as an exact surd and as a decimal; the steps show the squares being added (or subtracted for a leg).
- To test three lengths, choose “Check a triangle”; for a box or room, “Box diagonal”; for whole-number triples, “Pythagorean triples”.
- Choose the unit and decimal places, then copy or download the working — or download the list of triples as CSV.
Examples
a = 3, b = 4
c = 5
c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5.
a = 2, b = 3
c = √13 ≈ 3.6056
2² + 3² = 13, which is not a perfect square, so c stays √13.
a = 7, c = 10
b = √51 ≈ 7.1414
b² = c² − a² = 100 − 49 = 51.
6, 8, 10 · 4, 5, 6 · 2, 3, 4
right · acute · obtuse
10² = 100 = 36 + 64; 6² = 36 < 16 + 25; 4² = 16 > 4 + 9.
2 × 3 × 6
D = 7
Floor diagonal d² = 2² + 3² = 13; then D² = 13 + 6² = 49.
m = 5, n = 2
21, 20, 29
m² − n² = 21, 2mn = 20, m² + n² = 29, and 21² + 20² = 841 = 29².
Common uses
- Checking right-triangle homework, with exact surd answers as textbooks give them.
- Finding the length of a ladder, ramp, roof rafter or diagonal brace from the rise and the run.
- Testing whether a corner is square on site with the 3-4-5 rule (or 6-8-10 for larger corners).
- Working out whether a long object fits in a box, car boot or room (the space diagonal).
The theorem and its converse
In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c² (Euclid, Elements I.47). The converse is also true (I.48): if three lengths satisfy a² + b² = c², the angle opposite c is a right angle. Comparing c² with a² + b² for the longest side c tells you more: if c² is smaller the triangle is acute, if it is larger the triangle is obtuse. For triangles that are not right-angled, the law of cosines c² = a² + b² − 2ab cos C takes over.
Exact answers and simplified surds
A square root is simplified by taking out square factors: √52 = √(4 × 13) = 2√13, and √(13/36) = √13/6. The calculator does this exactly whenever the sides you type are whole numbers, decimals, fractions or square roots — decimals such as 1.5 are read as 3/2, so 1.5² + 2² = 6.25 gives exactly 2.5. In “Check a triangle” the comparison of c² with a² + b² is also exact, so 0.3, 0.4, 0.5 is reported as right-angled while 1, 1, 1.414 is (just) acute.
Pythagorean triples and Euclid’s formula
A Pythagorean triple is three whole numbers with a² + b² = c², such as 3, 4, 5. It is primitive when the three have no common factor. Euclid’s formula a = m² − n², b = 2mn, c = m² + n² (Elements X, Lemma 1 before Proposition 29) gives every primitive triple exactly once when m > n ≥ 1 have no common factor and one of them is even; every other triple is a multiple k of a primitive one. There are 16 primitive triples with a hypotenuse up to 100, and 52 triples in all.
Limitations
- The theorem holds for right triangles in flat (Euclidean) geometry. For long distances on the Earth’s surface, use a great-circle formula instead.
- Lengths must be positive and at most 10¹² in the chosen unit; decimals are rounded only for display.
- Lists of triples go up to a hypotenuse of 5,000, and the search for a given number up to 1,000,000.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
How do I find the hypotenuse of a right triangle?
Square the two legs, add them and take the square root: c = √(a² + b²). For legs 5 and 12, c = √(25 + 144) = √169 = 13. If the sum is not a perfect square, the exact answer is a surd: legs 2 and 3 give √13 ≈ 3.6056.
How do I find a missing leg?
Subtract instead of adding: a = √(c² − b²). With a hypotenuse of 13 and a leg of 12, the other leg is √(169 − 144) = √25 = 5. The hypotenuse must be the longest side, otherwise there is no such triangle.
How can I tell whether a triangle has a right angle?
Square the longest side and compare it with the sum of the squares of the other two. Equal means a right angle (6, 8, 10: 100 = 36 + 64). If the longest side’s square is smaller the triangle is acute; if larger, obtuse. This is the converse of the Pythagorean theorem.
What is the 3-4-5 rule?
Builders check that a corner is square by measuring 3 units along one side and 4 along the other: if the diagonal between the marks is exactly 5, the corner is 90°, because 3² + 4² = 5². Any multiple works too — 6-8-10 or 30-40-50 cm for bigger corners.
What are primitive Pythagorean triples?
Triples whose three numbers share no common factor, such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Triples such as 6-8-10 are multiples of a primitive one. In a primitive triple one leg is even and the other leg and the hypotenuse are odd.
Does a² + b² = c² work for any triangle?
Only for right triangles, with c the side opposite the right angle. For other triangles use the law of cosines, c² = a² + b² − 2ab cos C, which reduces to Pythagoras when C = 90° because cos 90° = 0.