Hypotenuse

Computed from default inputs using graph `synth-pythagorean-theorem`.

For a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c = √(a² + b²). Two conditions are load-bearing and both get violated in practice: the triangle must contain a true 90° angle, and c must be the side opposite it — always the longest side. Given the hypotenuse and one leg, the rearrangement flips the sign: a = √(c² − b²).

Solve for Leg a

Leg a: 3.7704 for Hypotenuse 5.5

Right-triangle hypotenuse formula.

hypotenuse=\sqrt{ legA\cdot legA+ legB\cdot legB}

hypotenusesqrt(legA * legA + legB * legB)

Hypotenuse decision comparison

item	value	note
Stress low	4	Leg a: 0
Conservative	4.59	Leg a: 2.25
Downside	4.83	Leg a: 2.7
Base case	5	Leg a: 3
Upside	5.19	Leg a: 3.3000000000000003

Visual Analysis1 series8 points

Expanded chart built from the same sensitivity run as the table, so users can read the trend and verify the numbers in one place.

Trend

Upward

Min

Max

4.5894

Insight

Hypotenuse shows an upward pattern, with a visible peak around 2.25 at 4.5894.

Visual response curve for Hypotenuse.

Will the couch fit through the stairwell turn?
A stair opening 42 in wide with 80 in of headroom offers a diagonal of √(42² + 80²) = 90.3 in. A 96 in couch cannot pass upright or flat — but the diagonal answer says an 88 in couch clears with margin, which no amount of measuring the width alone reveals. The theorem's daily job is exactly this: converting two perpendicular limits into the real constraint.

How does 3-4-5 square a real corner?
Mark 3 units along one edge and 4 along the other from the corner; the diagonal between the marks measures exactly 5 only when the corner is 90°. Longer multiples (6-8-10, 30-40-50 cm) improve accuracy because tape-reading error shrinks relative to the lengths. Deck framers, foundation layouts, and soccer-pitch lines all run on this one triple.

What about diagonals of rectangles and screens?
A rectangle's diagonal is the hypotenuse of its sides: a 16:9 screen advertised at 27 inches diagonal has width 23.5 in and height 13.2 in via this theorem. The reverse use — checking that a assembled frame's two diagonals measure equal — is the standard squareness test, again the theorem in disguise.

Does it extend beyond two dimensions?
Directly: the space diagonal of a box is √(l² + w² + h²) — apply the theorem twice. A 40 × 30 × 20 cm carton has a 53.9 cm internal diagonal, the longest rigid object it can hold. Each added dimension just adds another squared term.

Calculate the hypotenuse of a right triangle.

Pythagorean Theorem Calculator turns Leg a and Leg b into Hypotenuse. Confirm the units and assumptions match your situation before relying on the result.

Confirm the triangle has a right angle. The relation a² + b² = c² holds only when one angle is exactly 90°. Wall-to-floor, frame corners, and ladder-against-wall situations qualify; sloped or skewed layouts need the law of cosines instead.
Label the hypotenuse. The hypotenuse c is the side opposite the right angle and is always the longest side. The other two sides are the legs a and b.
Enter the two known sides in one unit. Convert everything to the same unit first — a run in feet and a rise in inches must both become feet (or inches) before squaring.
Add or subtract, then take the square root. Finding the hypotenuse: c = √(a² + b²). Finding a leg: a = √(c² − b²) — subtract, because the hypotenuse is already the largest side.
Sanity-check the result. The hypotenuse must be longer than either leg but shorter than their sum. A computed hypotenuse of 17 ft from legs of 12 ft and 5 ft (their plain sum) means the root was skipped.

Academic performance optimization

Important: Part-time work exceeding 20 hours/week is correlated with GPA decline (BLS data).

Enter your values, then check the formula, scenarios, and FAQ to confirm the assumptions fit your case. For medical, financial, legal, or safety-critical decisions, treat the output as an estimate and verify it against the relevant professional standard.

When is this mathematical calculation context enough?
Use it when the result only depends on base value, comparison value, rate and no local rule changes the interpretation.

What should I record with the result?
Record the input values, base definition, rounding, order of operations, output unit, and rounding rule used in the calculation.

When should I use a specialized modifier?
Use a specialized modifier when base value, rounding precision, inverse formula, and interpretation wording materially changes how the same result should be explained.

Applying it to a triangle that is not right-angled. On a 100-foot property line meeting another at 85°, the "hypotenuse" computed here is wrong by several feet. Non-right triangles need the law of cosines, which reduces to this theorem only when the angle is exactly 90°.

Adding instead of subtracting when solving for a leg. Given a 10 ft ladder (hypotenuse) reaching a wall with its foot 3 ft out, the height is √(100 − 9) = 9.54 ft — not √109. If the computed leg comes out longer than the hypotenuse, the signs were flipped; the hypotenuse is always longest.

Pythagorean Theorem Calculator

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Primary output (Hypotenuse)

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