Pythagorean Theorem Calculator

Choose whether to find hypotenuse c, find side a, find side b, or check three entered side lengths.

Enter the known positive lengths. The two legs a and b meet at the right angle, while c is the hypotenuse opposite the right angle.

Choose an optional unit label and the number of decimal places to display.

The result includes the requested side or verification, a compact calculation, a proportional diagram, and additional right-triangle measurements.

For a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

The relationship is written a² + b² = c², where c is always opposite the 90-degree angle.

Because c² contains both positive squared legs, c must be longer than either individual leg.

The equation connects side lengths only when a and b are perpendicular.

When both legs are known, add their squares and take the positive square root.

For legs 3 and 4, c = √(3² + 4²) = √25 = 5.

Only the positive square root is used because a geometric length cannot be negative.

When the hypotenuse and one leg are known, rearrange the theorem before taking the square root.

To find a, use a = √(c² − b²). To find b, use b = √(c² − a²).

The entered hypotenuse must be longer than the known leg. Otherwise the subtraction cannot produce a positive side length.

For b = 12 and c = 13, a = √(169 − 144) = √25 = 5.

The converse of the theorem allows three side lengths to be tested.

Place the longest side in c and compare a² + b² with c².

When they are equal within floating-point tolerance, the calculator reports a right triangle.

When they differ, the calculator shows the hypotenuse required by the two entered legs.

The tool first checks the triangle inequality, because three lengths that cannot form any triangle should not be treated merely as a non-right triangle.

Some side lengths are whole numbers, while others are irrational.

For legs 1 and 1, the exact hypotenuse is √2. The displayed decimal 1.4142 is an approximation.

When the squared expression is a supported safe integer, the calculator simplifies square factors. For example, √8 becomes 2√2.

Keep the radical form during symbolic work when possible and round the decimal only when a measured result is needed.

A Pythagorean triple is a set of positive whole numbers that satisfies a² + b² = c² exactly.

The best-known example is 3–4–5.

Multiplying every side by the same positive number preserves the proportions and creates another right triangle, such as 6–8–10.

Triples are convenient for layout checks and examples, but right-triangle sides do not need to be integers.

After all three sides are known, the calculator also derives the triangle’s area, perimeter, and two acute angles.

Area is ab ÷ 2 because either leg can act as the base and the other as its perpendicular height.

Perimeter is a + b + c.

The acute angle opposite a is calculated with atan(a ÷ b), and the angle opposite b uses atan(b ÷ a). Together with the 90-degree angle, they total 180 degrees.

The theorem applies whenever two perpendicular measurements determine a direct diagonal.

Common examples include rectangular frames, screens, ramps, ladders, roofs, coordinate distances, vector magnitudes, and square-corner checks.

The right angle must come from the geometry of the problem rather than being assumed merely because three lengths are available.

On a flat coordinate plane, the horizontal change Δx and vertical change Δy are perpendicular.

They form the legs of a right triangle whose hypotenuse is the straight-line distance between the points.

This gives d = √((x₂ − x₁)² + (y₂ − y₁)²), which is the Pythagorean theorem written with coordinate differences.

Measure three equal units along one edge and four along the other.

When the distance between those marks is five of the same units, the measurements form a 3–4–5 right triangle.

Larger scaled versions such as 6–8–10 can be easier to measure accurately.

Real layout work still requires suitable tolerances, tools, reference lines, and safety practices.

Every side must use the same unit before the theorem is applied.

Do not enter one side in feet and another in inches unless one is converted first.

The unit selector only labels the answer. It does not alter the numerical values.

Multiplying every side by the same positive scale factor creates a similar right triangle. Lengths scale linearly, perimeter scales linearly, and area scales by the square of the factor.

The formulas can be exact while the entered measurements are approximate.

A tape measure, drawing, coordinate reading, or instrument may already include rounding and uncertainty.

Choose decimal precision that matches the source measurements rather than presenting unsupported extra digits.

Avoid rounding an intermediate side before using it in another area, angle, or construction calculation.

The calculator uses Math.hypot when finding the hypotenuse so large side values do not need to be squared directly.

Missing-leg calculations use a factored form of c² − b² or c² − a² to reduce cancellation and overflow problems.

Check mode scales all three sides before comparing their squares.

Extremely large results that remain outside JavaScript’s finite numeric range are rejected or marked unavailable.

The right-triangle relationship was known in ancient mathematical traditions before the surviving Greek treatments.

Euclid presented a geometric proof in Book I, Proposition 47 of the Elements.

The familiar name associates the theorem with Pythagoras, but the historical record does not establish that one person was solely responsible for discovering every form of the relationship.

The equation a² + b² = c² applies only to a right triangle.

A triangle without a 90-degree angle generally requires the law of cosines, law of sines, or additional angle information.

The two-dimensional theorem does not directly calculate a three-dimensional box diagonal unless it is applied in two stages.

Long routes over Earth require geographic or geodesic distance methods rather than a flat-plane triangle.

Structural ramps, roofs, ladders, braces, and surveying work may require tolerances, loading, code compliance, material thickness, sag, terrain, and professional review beyond this calculation.

All side parsing, calculation, verification, formatting, and diagram generation run in the browser.

Entered values are not uploaded, added to a public URL, or permanently stored.

Clipboard access is requested only when the primary result is copied.