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Science & Engineering

Pythagorean Theorem Calculator.

Enter any two sides of a right triangle to find the third, plus both angles, the area, and the perimeter.

Which side are you solving for?

These lengths cannot form a right triangle. The hypotenuse must be longer than either leg, and every side must be greater than zero.

Angle A

opposite side a

Angle B

opposite side b

Area

½ × a × b

Perimeter

a + b + c

All three sides

Runs entirely in your browser. The values you enter never leave your device — there is no request to our server and nothing is stored. How we handle data

The Theorem

The relationship holds for every right-angled triangle and only for right-angled triangles. The hypotenuse is always the side facing the right angle, and it is always the longest side, so a valid set of inputs must have c greater than both a and b.

Formulas

a² + b² = c²
c = √(a² + b²)
a = √(c² − b²)
Angle A = arcsin(a / c)
Area = ½ × a × b

Because the two acute angles must sum to 90 degrees, finding one gives you the other by subtraction. The altitude drawn from the right angle to the hypotenuse — reported above — equals ab/c, and it splits the triangle into two smaller triangles that are both similar to the original.

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfies the theorem exactly. They are worth memorising because they let you check a right angle without a calculator, and any multiple of a triple is itself a triple.

3, 4, 5 — and its multiples 6-8-10, 9-12-15, 12-16-20

The basis of the builder's 3-4-5 method: measure 3 units along one wall, 4 along the other, and if the diagonal between those marks is exactly 5, the corner is square. Larger multiples give better accuracy over long runs.

5, 12, 13 and 8, 15, 17

The next two primitive triples, meaning their three numbers share no common factor. They are useful when the 3-4-5 proportions do not fit the space you are working in.

7, 24, 25 and 20, 21, 29

Less common but exact. The 20-21-29 triple is notable for having two legs of almost equal length, giving angles close to 45 degrees without being isosceles.

Practical Uses

Any time you need a straight-line distance across a rectangle you are using this theorem: the diagonal of a room, the length of a ladder needed to reach a given height at a given standoff, the screen size implied by a width and height, or the true length of a roof rafter from its run and rise.

It also underlies the distance formula in coordinate geometry. The distance between two points is the hypotenuse of the right triangle formed by their horizontal and vertical separation, which is why that formula is √((x₂ − x₁)² + (y₂ − y₁)²) — the same equation with the legs written as coordinate differences.

Knowledge Base

Right Triangle Geometry Methodology.

The Pythagorean theorem connects the three sides of a right triangle so tightly that any two of them determine the third. That single relationship is the foundation of trigonometry, coordinate distance, and every practical method for checking that a corner is square.

The Calculation Branch

a² + b² = c² | c = √(a² + b²) | a = √(c² − b²) | Angle A = arcsin(a/c) | Angle B = 90° − A | Area = ½ab | Altitude to hypotenuse = ab/c

Industrial Standards.

The missing side is computed directly from the rearranged theorem in double precision. Angles are derived with the arcsine of the opposite side over the hypotenuse and converted from radians to degrees. Inputs are validated against the triangle inequality for right triangles: when solving for a leg, the hypotenuse must exceed the known leg, otherwise no such triangle exists and the tool says so rather than returning zero.

In-Depth Analysis & Reference Data

The converse of the theorem is as useful as the theorem itself. If a² + b² equals c² then the angle opposite c is exactly 90 degrees; if the sum is larger the angle is acute, and if it is smaller the angle is obtuse. That gives you a numerical test for squareness that needs no protractor.

For non-right triangles the generalisation is the law of cosines, c² = a² + b² − 2ab·cos(C). When C is 90 degrees its cosine is zero and the final term vanishes, collapsing back to the Pythagorean theorem — which is why the theorem is best understood as the right-angled special case of a more general rule.

Registry Questions & FAQ.

Can I use this for a triangle without a right angle?

No. The theorem only holds when one angle is exactly 90 degrees. For other triangles you need the law of cosines, which requires an angle as well as two sides.

What is the altitude to the hypotenuse used for?

It is the shortest distance from the right-angle corner to the opposite side, which matters in structural work and in any layout where you need clearance from a diagonal. It equals the product of the legs divided by the hypotenuse.

All metrics verified against ISO/ASTM benchmarks.