Hypotenuse Calculator

Enter the two legs of a right triangle to find the hypotenuse (c = √(a² + b²)), the two acute angles, and the triangle's area.

Quick Facts

Pythagorean theorem
a² + b² = c²
The hypotenuse c is √(a² + b²), the longest side of a right triangle.
Angle relations
sin(α) = a/c, cos(α) = b/c
Once c is known, inverse trig functions give both acute angles.
Common triple
3-4-5
Legs 3 and 4 always give a hypotenuse of exactly 5.

Your Results

Calculated
Hypotenuse (c)
-
c = √(a² + b²)
Angle opposite a (α)
-
α = atan(a / b)
Angle opposite b (β)
-
β = 90° − α
Triangle area
-
Area = (a × b) / 2

Ready

Enter both legs and a unit, then press Calculate.

Formula and Method for Finding the Hypotenuse

The hypotenuse is the longest side of a right triangle — the one opposite the 90° angle. The Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides (the legs): a² + b² = c². Solving for c gives the hypotenuse formula: c = √(a² + b²). This calculator also derives the triangle's two acute angles and its area from the same two legs.

How the calculation works

Enter the lengths of the two legs, a and b, in any consistent unit. The calculator squares each leg, adds the squares together, and takes the square root to get the hypotenuse c. It then uses inverse trigonometry to find the acute angles: α = atan(a / b) is the angle opposite leg a, and β = 90° − α is the angle opposite leg b (the two acute angles in a right triangle always sum to 90°). Finally, the area of the right triangle is (a × b) / 2, since the two legs meet at a right angle and act as base and height.

Common mistakes

  • Applying the theorem to non-right triangles: a² + b² = c² only holds when the triangle has a genuine 90° angle between legs a and b. For other triangles, use the Law of Cosines instead.
  • Mixing up the hypotenuse with a leg: the hypotenuse is always the side opposite the right angle and is always the longest side — never substitute it for a or b in the formula.
  • Mixing units: keep both legs in the same unit (both in feet, both in meters, etc.) before calculating; convert first if your measurements come in different units.

Real-world applications

  • Construction and carpentry use the 3-4-5 rule (a Pythagorean triple) to square corners and verify that walls or foundations meet at exactly 90°.
  • Ladder and safety calculations use the hypotenuse to find how far a ladder's base should sit from a wall to reach a given height safely.
  • Navigation and surveying use right-triangle relationships to compute straight-line (as-the-crow-flies) distances from perpendicular east-west and north-south offsets.
  • Screen and display sizing (TVs, monitors) is quoted as the diagonal, which is the hypotenuse of the right triangle formed by the screen's width and height.

Frequently Asked Questions

What is the formula for finding the hypotenuse of a right triangle?
The hypotenuse equals the square root of the sum of the squares of the two legs: c = √(a² + b²). This comes directly from the Pythagorean theorem, a² + b² = c². For example, legs of 3 and 4 give a hypotenuse of √(9 + 16) = √25 = 5.
Why is the hypotenuse always the longest side of a right triangle?
The hypotenuse is the side opposite the 90° angle, which is the largest angle in a right triangle. In any triangle, the longest side is always opposite the largest angle, so the hypotenuse is necessarily longer than either leg.
How do I find a missing leg if I already know the hypotenuse and one leg?
Rearrange the Pythagorean theorem to solve for the unknown leg: b = √(c² − a²). This calculator solves for the hypotenuse from two legs, but the same equation works in reverse for a missing leg as long as c is greater than the known leg.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c² exactly, such as 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Any multiple of a known triple, like 6-8-10, is also a valid triple.

Related calculators