What is the Pythagorean theorem?

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². It was known to ancient Babylonian mathematicians around 1800 BCE and later formally proven by the Greek mathematician Pythagoras around 570–495 BCE. The theorem applies only to right triangles — those with one 90-degree angle.

How do you find the hypotenuse of a right triangle?

To find the hypotenuse, square both legs, add them together, and take the square root: c = √(a² + b²). For example, if a = 3 ft and b = 4 ft, then c = √(9 + 16) = √25 = 5 ft. This is the classic 3-4-5 right triangle. Always make sure both legs are in the same unit before calculating.

How do you find a missing leg of a right triangle?

To find a missing leg when you know the hypotenuse and one leg, rearrange the formula: b = √(c² − a²). For instance, if the hypotenuse c = 13 and one leg a = 5, then b = √(169 − 25) = √144 = 12. The hypotenuse must always be larger than either leg — if it is not, your inputs are incorrect.

What are common Pythagorean triples?

Pythagorean triples are whole-number sets that satisfy a² + b² = c². The most common are: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 20-21-29. Any multiple of these also works — 6-8-10, 9-12-15, etc. Carpenters and builders use the 3-4-5 triple constantly to check for square corners: measure 3 ft along one wall, 4 ft along the other, and the diagonal should be exactly 5 ft.

What are real-world uses of the Pythagorean theorem?

The Pythagorean theorem is used in construction (squaring foundations and walls), navigation (calculating straight-line distance between two points), surveying, architecture, and computer graphics. It is the basis of the distance formula in coordinate geometry: distance = √((x₂−x₁)² + (y₂−y₁)²). Electricians use it to calculate wire lengths across diagonal runs, and roofers use it to calculate rafter lengths from run and rise.

What is the derivation or proof of the Pythagorean theorem?

There are over 370 known proofs of the Pythagorean theorem. One simple geometric proof: arrange four identical right triangles inside a square to form a smaller square in the center. The area of the outer square equals (a + b)², and it also equals the four triangle areas (4 × ½ab) plus the inner square (c²). Setting these equal and simplifying gives a² + b² = c². U.S. President James Garfield published his own original proof in 1876.

Does the Pythagorean theorem work in 3D?

Yes. The 3D extension of the Pythagorean theorem gives the distance between two points in three dimensions: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). You can also find the space diagonal of a rectangular box with dimensions a × b × c using d = √(a² + b² + c²). This is simply the Pythagorean theorem applied twice: first to find the face diagonal, then to use that as a leg to find the space diagonal.

How is the 3-4-5 rule used in construction?

The 3-4-5 rule is the most practical application of the Pythagorean theorem in construction. To check if a corner is square (exactly 90°): measure 3 feet from the corner along one wall, 4 feet along the other wall, and verify the diagonal between those two points is exactly 5 feet. If it is, the corner is square. Larger multiples work too: 6-8-10 or 9-12-15. This technique is used to square building foundations, deck posts, tile layouts, and framing corners.

Pythagorean Theorem Calculator — Solve a² + b² = c²

How to Use the Pythagorean Theorem Calculator

This Pythagorean theorem calculator finds the missing side of any right triangle — select a mode (Find hypotenuse (c) or Find missing leg), then enter the known sides. The calculator instantly returns the missing side length along with imperial and metric equivalents. All inputs support unit switching between inches, feet, yards, meters, and centimeters, so you never need to convert manually. For problems involving three-dimensional space or vector geometry, see our cross product calculator.

The built-in right-triangle diagram labels all three sides (a, b, and c) so you always know which value you are entering. The result card also displays a unit equivalent — for example, 5.00 ft · 1.52 m — so metric and imperial users both get a useful reading.

The Pythagorean Theorem Formula

The Pythagorean theorem expresses a fundamental relationship between the three sides of every right triangle:

a² + b² = c²

Where a and b are the two legs (the sides that form the right angle) and c is the hypotenuse (the side opposite the right angle). The hypotenuse is always the longest side.

Finding the Hypotenuse

When you know both legs, solve for c by rearranging the formula: c = √(a² + b²). Example: a = 6 ft, b = 8 ft → c = √(36 + 64) = √100 = 10 ft. This is the 3-4-5 triple scaled by 2. When the result is a non-perfect square, our radical calculator simplifies radical expressions like √50 to 5√2. For the area of a right triangle formed by the same legs, see our triangle area calculator — area = ½ × a × b.

Finding a Missing Leg

When you know the hypotenuse and one leg, solve for the other: b = √(c² − a²). Example: c = 13 in, a = 5 in → b = √(169 − 25) = √144 = 12 in. The hypotenuse must be strictly larger than either leg — if it is not, the triangle cannot exist.

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Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) that exactly satisfy a² + b² = c². These are especially useful in construction and carpentry because they produce perfect right angles with no fractions or irrational numbers.

3-4-5 — the most common; used by builders to square foundations
5-12-13 — useful for longer spans; 5² + 12² = 25 + 144 = 169 = 13²
8-15-17 — common in structural applications
7-24-25 and 20-21-29 — less common but still whole-number triples

Any integer multiple of a triple also forms a right triangle: 6-8-10, 9-12-15, 10-24-26, and so on. To check if any three numbers form a right triangle, plug the two smaller values in as a and b — if a² + b² = c², you have a right triangle.

Real-World Applications of the Pythagorean Theorem

The Pythagorean theorem appears constantly in construction, engineering, navigation, and everyday DIY tasks. Here are some of the most common real-world uses:

Squaring a foundation or wall — measure 3 ft along one side, 4 ft along the other; the diagonal must be 5 ft for a true 90° corner
Calculating rafter length — rise and run are the two legs; the rafter is the hypotenuse. A roof with a 6-ft rise and 8-ft run needs a 10-ft rafter
Stair stringers — rise and run determine the diagonal stringer length
Navigation and GPS — straight-line distance between two coordinates uses the distance formula, which is the Pythagorean theorem in disguise
Screen sizes — TV and monitor sizes are the diagonal of the screen; width and height are the two legs. For a 16:9 screen that is 55 in diagonal, the width is about 47.9 in and height about 27.0 in

For volume problems involving cylindrical or three-dimensional shapes, our cubic feet calculator handles the geometry automatically.

Step-by-Step Example

Problem: A ladder leans against a wall. The base of the ladder is 6 feet from the wall and the ladder reaches 8 feet up the wall. How long is the ladder?

Identify the legs: a = 6 ft (horizontal distance), b = 8 ft (vertical height)
Apply the formula: c = √(a² + b²) = √(36 + 64) = √100
Result: c = 10 ft — the ladder is 10 feet long

Verification: 6² + 8² = 36 + 64 = 100 = 10². Confirmed.

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Common Mistakes

Using mixed units — always convert all sides to the same unit before calculating. 3 feet and 48 inches are both 3 ft, but only if you convert first
Confusing legs and hypotenuse — the hypotenuse is always opposite the right angle and is always the longest side. Never substitute a leg where the hypotenuse is expected
Applying to non-right triangles — the theorem only works for triangles with exactly one 90° angle. For other triangles, use the law of cosines
Forgetting to take the square root — a² + b² gives c², not c. You must take the square root of the sum to get the side length
Rounding intermediate steps — always carry full precision through the calculation and round only the final answer to avoid compounding errors

Sources & References

Pythagorean Theorem — Khan Academy
Pythagorean Theorem — Britannica — Encyclopædia Britannica

Pythagorean Theorem Calculator