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Pythagorean Theorem Calculator

Find any side of a right triangle — or check if a triangle is right, acute, or obtuse.

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Result
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Common Pythagorean Triples

abc (hypotenuse)Verification (a²+b²=c²)Notes
3459+16=25 ✓Most common
5121325+144=169 ✓Used in construction
8151764+225=289 ✓Classic triple
7242549+576=625 ✓Integer triple
202129400+441=841 ✓Almost-isosceles
9404181+1600=1681 ✓Large gap
123537144+1225=1369 ✓—
116061121+3600=3721 ✓—
681036+64=100 ✓Multiple of 3-4-5
9121581+144=225 ✓Multiple of 3-4-5

Construction Diagonal

To check if a room's corners are square (90°), measure the diagonal. For a 12×16 ft room: diagonal = √(144+256) = √400 = 20 ft. If measured diagonal = 20 ft, corners are square.

TV / Screen Size

A "55-inch TV" means the diagonal is 55 inches. For a 16:9 ratio: width = 55×16/√(16²+9²) ≈ 47.9 in, height ≈ 26.9 in. The Pythagorean theorem defines that diagonal.

Ramp / Slope Distance

A wheelchair ramp rises 2 ft over a horizontal run of 24 ft. The actual ramp length = √(2²+24²) = √(4+576) = √580 ≈ 24.08 ft — you need slightly more than the horizontal length.

Navigation & Distance

If you travel 30 miles east and 40 miles north, your straight-line distance from start = √(30²+40²) = √(900+1600) = √2500 = 50 miles — the "as the crow flies" distance.

How to Use This Calculator

1

Enter Two Known Sides

Provide the lengths of any two sides of a right triangle (a, b, or c).

2

Select the Unknown Side

Indicate which side you want to solve for—the hypotenuse or a leg.

3

Get the Missing Side

The calculator applies the Pythagorean theorem and displays the result with steps.

Formula & Methodology

Pythagorean Theorem

a² + b² = c²

The sum of the squares of the two legs equals the square of the hypotenuse.

Solve for Hypotenuse

c = √(a² + b²)

Take the square root of the sum of the squared legs to find the hypotenuse.

Solve for a Leg

a = √(c² − b²)

Subtract the square of the known leg from the square of the hypotenuse, then take the square root.

About this calculation · Accuracy & Methodology

Key Terms

Hypotenuse (c)
The longest side of a right triangle, opposite the right angle.
Leg (a or b)
Either of the two shorter sides of a right triangle that form the right angle.
Right Angle
An angle of exactly 90 degrees, forming an L-shape.
Pythagorean Triple
A set of three positive integers (a, b, c) that satisfy a² + b² = c², such as (3, 4, 5).
Right Triangle
A triangle containing exactly one 90-degree angle.

Real-World Examples

Example 1

Ladder Against Wall

a = 6 ft, b = 8 ft

c = 10 ft — a 6-8-10 right triangle (scaled 3-4-5 triple)

Example 2

TV Screen Diagonal

a = 30 in, c = 55 in

b = 46.10 in — finding the height of a 55-inch widescreen TV

Common Pythagorean Triples

Leg aLeg bHypotenuse cScale Factor
3451×
512131×
815171×
724251×
68102× (3-4-5)

The Pythagorean Theorem: Cornerstone of Geometry

A Theorem with 400+ Proofs

The Pythagorean theorem, attributed to the Greek mathematician Pythagoras around 500 BCE, is one of the most proven results in mathematics. Over 400 distinct proofs exist, including geometric, algebraic, and even a proof by US President James Garfield. The theorem holds in any Euclidean plane and generalizes to higher dimensions.

Real-World Applications

Builders use the 3-4-5 triple to verify right angles on construction sites. Navigation systems compute straight-line distances using the theorem. In computer graphics, collision detection between objects relies on distance calculations rooted in the Pythagorean theorem. Its simplicity and universality make it one of the most practically useful results in all of mathematics.

Beyond Euclidean Geometry

On curved surfaces like a sphere, the Pythagorean theorem does not hold exactly. Spherical geometry modifies the relationship between sides and angles, which is why long-distance navigation uses the Haversine formula instead. In Einstein's theory of relativity, a modified version of the theorem applies to spacetime intervals.