Geometry
Pythagorean Theorem Calculator
Find the hypotenuse or a missing leg of a right triangle with a² + b² = c². A live diagram, perimeter, area and angles update as you type — with every step of the arithmetic shown.
Solve for
Enter the two legs (the sides that meet at the right angle).
Hypotenuse c
5
Whole-number Pythagorean triple 3-4-5
Perimeter
12 cm
Area
6 cm²
Angle ∠A
36.87°
Angle ∠B
53.13°
Try an example
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The basics
What is the Pythagorean theorem?
The Pythagorean theorem ties together the three sides of a right triangle. The two shorter sides that meet at the right angle are the legs; the long side opposite the right angle is the hypotenuse. Square the legs, add them, and you get the square of the hypotenuse — a relationship that has held for every right triangle for over two thousand years.
a² + b² = c²
The two legs and the hypotenuse
The squares built on the two legs a and b together equal the square built on the hypotenuse c. It is true only when one angle is exactly a right angle.
3² + 4² = 9 + 16 = 25 = 5²
c = √(a² + b²)
Why the hypotenuse is longest
Because you add two positive squares, the hypotenuse always exceeds either leg on its own. The longest side sits opposite the largest angle, and a right angle is the largest in a right triangle.
√(9 + 16) = √25 = 5
Method
How to solve it by hand
The same three moves work whether you are after the hypotenuse or a missing leg — only the rearrangement changes.
- 1
Identify what you have and what you want
Mark which sides are legs and which is the hypotenuse. If you know both legs you are solving for c; if you know the hypotenuse and one leg you are solving for the other leg.
legs → c · c and a leg → leg - 2
Write the formula and rearrange
For the hypotenuse use it as written. For a leg, move the known leg to the other side so the unknown is alone.
c = √(a² + b²) · b = √(c² − a²) - 3
Substitute the numbers
Drop your values in, square them, then add or subtract as the formula demands.
√(3² + 4²) = √(9 + 16) = √25 - 4
Take the square root
The square root of the result is the side length. Round only at the very end.
√25 = 5
Worked examples
See it across cases
The procedure is identical whether the answer is a clean whole number or an irrational decimal. Switch tabs to watch it play out.
Two legs given — solve for c.
Given a = 6, b = 8
c² = a² + b² = 6² + 8²
c² = 36 + 64 = 100
c = √100
c = 10
Quick chart
Pythagorean triples list
Triples are right triangles with whole-number sides — handy to recognise so you can skip the square roots. These are the ones that appear most often in homework and on the job.
3, 4, 5
3²+4²=5²5, 12, 13
5²+12²=13²8, 15, 17
8²+15²=17²7, 24, 25
7²+24²=25²20, 21, 29
20²+21²=29²9, 40, 41
9²+40²=41²12, 35, 37
12²+35²=37²11, 60, 61
11²+60²=61²28, 45, 53
28²+45²=53²16, 63, 65
16²+63²=65²33, 56, 65
33²+56²=65²48, 55, 73
48²+55²=73²Primitive triples share no common factor. Multiply any of them by a whole number and the result is still a valid right triangle.
Where it shows up
Why it matters
One of the most-used formulas in all of mathematics, quietly at work far beyond the classroom.
3-4-5
Construction
Carpenters square up corners with the 3-4-5 rule — measure three units along one wall, four along the next, and the diagonal reads five when the corner is a true right angle.
√(Δx² + Δy²)
Navigation & maps
Straight-line (“as the crow flies”) distance between two map points is the hypotenuse of the east-west and north-south gaps.
a² + b² = c²
Screens & displays
A TV’s advertised size is its diagonal — the hypotenuse computed from its width and height, which is why aspect ratio changes the usable area.
Connections
Formulas built on the theorem
Once you know a² + b² = c², several other staples fall straight out of it.
d = √((x₂−x₁)² + (y₂−y₁)²)
The distance formula
The gap between two points is the hypotenuse of the horizontal and vertical differences — the theorem dressed up in coordinates.
c² = a² + b² − 2ab·cos C
The law of cosines
The general triangle rule. When angle C is 90° its cosine is zero and the whole thing collapses back to the Pythagorean theorem.
√(x² + y² + z²)
3D space diagonals
Apply the theorem twice — once in the base plane, once up the height — to get the diagonal across a box or room.
r² = x² + y²
The equation of a circle
Every point on a circle of radius r sits a hypotenuse-length away from the centre, which is why the circle equation is Pythagorean at heart.
Worth knowing
Rules and pitfalls
A handful of facts that keep you from the classic mistakes.
c is always longest
The hypotenuse can't be a leg
Always pin down which side is the hypotenuse first — it is the one opposite the right angle, and it is always the longest. Solving for a leg but plugging into the hypotenuse slot is the most common error.
√(a² + b²) ≠ a + b
You can't add before squaring
The square root of a sum is not the sum of the roots. You must square, then add, then root — in that order — or the answer comes out far too large.
a² + b² = c² ⇒ right angle
The converse is a test
The theorem runs both ways. If the three sides satisfy the equation, the triangle must contain a right angle — which is exactly how the 3-4-5 builder's trick works.
acute vs obtuse
It also classifies triangles
If a² + b² is greater than c² the triangle is acute; if it is less, the triangle is obtuse. Equality is the knife-edge right angle.
units stay consistent
Mix units and the answer breaks
All three sides must be in the same unit before you compute. Convert first; the calculator keeps a single unit across every field for exactly this reason.
round last
Round only at the end
Squaring magnifies rounding error. Carry full precision through the arithmetic and round the final side length, not the intermediate squares.
Origin
A very old result
The relationship is named for the Greek mathematician Pythagoras (c. 570–495 BC), whose school is credited with the first known proof. But the rule was used far earlier: Babylonian clay tablets list Pythagorean triples a thousand years before him, and ancient Egyptian and Indian builders used knotted 3-4-5 ropes to lay out right angles. Today there are hundreds of distinct proofs — geometric, algebraic, even one attributed to a US president — making it one of the most-proven theorems in mathematics.
FAQ
Common questions
For any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse — the side opposite the 90° angle. It holds only for right triangles.
Square both legs, add them, and take the square root: c = √(a² + b²). For legs of 3 and 4, c = √(9 + 16) = √25 = 5.
Rearrange the formula to b = √(c² − a²). Square the hypotenuse, subtract the square of the known leg, then take the square root. The hypotenuse must be longer than the known leg or no right triangle exists.
A Pythagorean triple is a set of three whole numbers a, b and c that satisfy a² + b² = c² — for example 3-4-5, 5-12-13 and 8-15-17. They describe right triangles whose sides are all integers.
No. It works only for right triangles, which contain a 90° angle. For triangles without a right angle, use the law of cosines: c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when C = 90°.
The 3-4-5 rule uses the smallest Pythagorean triple to check for a square (90°) corner. If a triangle measures 3 and 4 along two sides and exactly 5 across, the corner between the short sides is a perfect right angle — builders use it to square up walls and foundations.
The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²). That is just the Pythagorean theorem applied to the horizontal and vertical gaps as the two legs of a right triangle.
No. The hypotenuse is always the longest side of a right triangle because it sits opposite the largest angle (90°). If you are told the hypotenuse is shorter than a leg, the figure is not a valid right triangle.
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Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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