Hypotenuse
c² = a² + b² ⇒ c = √(a² + b²)
Add the squares of the legs, then take the square root.
Pythagorean Theorem Calculator with Converse and Steps
Calculate the hypotenuse or a leg of a right triangle with the Pythagorean theorem, test whether a triangle is right-angled with the converse and contrapositive, and get the step-by-step working: equation, squares, square root and conclusion.
- Clipboard
Result
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Enter two positive lengths. To find a leg, the hypotenuse must be longer than the other leg.
Enter three positive lengths that form a triangle: the longest side must be shorter than the sum of the other two.
Worked solution
- Enter two lengths to see the full worked solution.
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How does this tool work?
This calculator uses the Pythagorean theorem (Pythagoras’ theorem) to find the hypotenuse or a missing leg of a right triangle, and its converse to tell whether a triangle with three known sides is right-angled. Each answer comes with the working you would write in class: the equation a² + b² = c², the substituted values, the squares, the exact square root and a rounded value.
The Pythagorean theorem states: if triangle ABC has a right angle at A, the square of the hypotenuse equals the sum of the squares of the other two sides, BC² = AB² + AC². The hypotenuse BC is the side opposite the right angle and always the longest side; the other two sides are called the legs.
To find the hypotenuse, add the squares of the legs and take the square root: with AB = 3 and AC = 4, BC² = 9 + 16 = 25, so BC = √25 = 5. To find a leg, rearrange by subtracting: AB² = BC² − AC². When the square root is not a whole number, the calculator keeps the exact value (for example √52) and also gives an approximation to two decimal places.
The converse proves that a triangle is right-angled: if the square of the longest side equals the sum of the squares of the other two sides, the triangle has a right angle, located at the vertex opposite the longest side. The calculator finds the longest side for you before comparing.
The contrapositive proves the opposite: if the square of the longest side is not equal to the sum of the other two squares, the triangle is not a right triangle. Comparing c² with a² + b² also tells you more: if c² is smaller, every angle is acute; if it is larger, the angle opposite c is obtuse.
A complete written answer states the right angle, names the theorem, writes the equation with letters, substitutes the values, works out the squares and the square root, and ends with a sentence that includes the unit. For the converse, work out the square of the longest side and the sum of the other two squares separately, compare them, then quote the converse or the contrapositive.
How to use this tool
Pick a mode
Theorem to find a length in a right triangle; Converse to check whether a triangle is right-angled.
Choose the side to find and the unit
The hypotenuse BC or one of the legs, AB or AC; the fields adapt to your choice.
Enter the known lengths
Two lengths in Theorem mode, all three sides in Converse mode; the result updates as you type.
Read and copy the solution
Equation, squares, square root and conclusion are laid out as in a notebook; the button copies them as plain text.
Pythagorean theorem formulas
In a right triangle, a and b are the legs and c is the hypotenuse.
Missing leg
a² = c² − b² ⇒ a = √(c² − b²)
Subtract the square of the other leg from the square of the hypotenuse.
Converse
c² = a² + b² ⇒ right triangle
Here c is the longest side; the right angle is at the vertex opposite c.
Contrapositive
c² ≠ a² + b² ⇒ not a right triangle
If the equation fails for the longest side, the triangle has no right angle.
- a, b
- legs, the two sides that form the right angle
- c
- hypotenuse, opposite the right angle and the longest side
Example: with a = 3 and b = 4, c = √(3² + 4²) = √25 = 5.
Use cases
Homework and exam practice
Check a hypotenuse calculation or a right-angle proof and compare your working with a complete solution.
DIY and construction
Square a corner with the 3-4-5 rule (3 ft, 4 ft, 5 ft or 60 cm, 80 cm, 1 m), or work out the length of a rafter, ramp or ladder.
Diagonals and distances
Find the diagonal of a screen, a room or any rectangle, or a straight-line distance on a grid.
Exact values and rounding
Practise keeping an exact square root and giving a correctly rounded approximation.
Examples
Find the hypotenuse
BC² = 3² + 4² = 9 + 16 = 25, so BC = √25 = 5 cm.
Input
AB = 3 cm, AC = 4 cmResult
BC = 5 cmHypotenuse that is not a whole number
BC² = 16 + 36 = 52; √52 is not a whole number, so it is rounded to 7.21 cm.
Input
AB = 4 cm, AC = 6 cmResult
BC = √52 ≈ 7.21 cmFind a leg
AB² = BC² − AC² = 169 − 25 = 144, so AB = √144 = 12 cm.
Input
BC = 13 cm, AC = 5 cmResult
AB = 12 cmHow high a ladder reaches
h² = 5² − 1.4² = 25 − 1.96 = 23.04, so h = √23.04 = 4.8 m.
Input
5 m ladder, foot 1.4 m from the wallResult
height = 4.8 mDiagonal of a rectangle
d² = 1.44 + 0.25 = 1.69, so d = √1.69 = 1.3 m.
Input
sides of 1.2 m and 0.5 mResult
diagonal = 1.3 mConverse: right triangle
BC² = 100 and AB² + AC² = 36 + 64 = 100, so the equation holds.
Input
AB = 6, AC = 8, BC = 10Result
right angle at AContrapositive: not a right triangle
8² = 64 but 5² + 6² = 25 + 36 = 61, and 64 ≠ 61.
Input
sides of 5, 6 and 8Result
not a right triangleTips and best practices
- Spot the hypotenuse first: it is opposite the right angle and is the longest side. It stands alone on one side of BC² = AB² + AC².
- To find a leg, subtract: AB² = BC² − AC². Adding the squares instead is the most common mistake.
- Do not forget the final square root: BC² = 25 means BC = 5, not 25.
- Keep the exact value (√52) until the end and round only at the last step, to the accuracy you are asked for.
- For the converse, compare the square of the longest side with the sum of the other two squares, never an arbitrary side.
- Use one unit throughout: convert centimetres and metres (or inches and feet) before squaring.
Sources and references
- Weisstein E.W., Pythagorean Theorem (opens in a new tab), Wolfram MathWorldPage in English
- Euclid, ed. D.E. Joyce, Elements, Book I, Proposition 47 (opens in a new tab), Clark UniversityPage in English
- Euclid, ed. D.E. Joyce, Elements, Book I, Proposition 48 (opens in a new tab), Clark UniversityPage in English
- Programme de mathématiques pour le cycle 4 (annexe 2) (opens in a new tab), Ministère de l’Éducation nationalePage in French
Frequently asked questions
When can I use the Pythagorean theorem?
Only in a right triangle, when you know two sides and need the third. If the triangle has no right angle, the equation does not apply and you need other tools, such as trigonometry or the law of cosines.
How do I find the hypotenuse?
Add the squares of the two legs and take the square root: c = √(a² + b²). For legs of 6 and 8 cm, √(36 + 64) = √100 = 10 cm.
How do I find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² − b²). With c = 13 and b = 5, √(169 − 25) = √144 = 12.
What are the converse and the contrapositive?
The theorem starts from a right triangle to find a length. The converse starts from c² = a² + b², with c the longest side, to prove that a triangle is right-angled. The contrapositive starts from c² ≠ a² + b² to prove that it is not.
Why does the calculator show a square root and a decimal?
When the result is not a perfect square, the exact length is written with a square root, such as √52. The calculator keeps it and also gives the rounded value, 7.21, which is what you need for a real measurement.
What are Pythagorean triples?
Sets of three whole numbers that satisfy a² + b² = c², such as 3, 4, 5 or 5, 12, 13. Any multiple of a triple is also a triple (6, 8, 10), which is why builders use the 3-4-5 rule to set out a right angle.
Does the theorem work in three dimensions?
Yes, by applying it twice. The space diagonal of a box with sides a, b and c is √(a² + b² + c²): first find the diagonal of the base, then use it as a leg of a second right triangle.
Is my data sent to a server?
No. The calculations and the worked solution run entirely in your browser, with no account and no data sent anywhere.
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