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Math

Calculate the Hypotenuse of a Triangle

Updated · Standard formula, shown below

Hypotenuse c 5.000 cm
Angle opposite leg a
36.87 degrees
Result
Area
6.00 cm²
Result
Perimeter
12.000 cm
Result

This calculator finds the hypotenuse of a right triangle from the lengths of the two shorter sides (the legs). Enter both legs in the same unit and you get the hypotenuse, the angle opposite side a, the area and the perimeter.

Formula

The Pythagorean theorem: the square of the hypotenuse equals the sum of the squares of the legs, c² = a² + b², so c = √(a² + b²). The angle opposite leg a is A = arctan(a ÷ b). Area = a × b ÷ 2, perimeter = a + b + c.

  • a, b: the two legs that meet at the right angle (any length unit, both the same)
  • c: the hypotenuse, the side opposite the right angle (same unit)
  • A: the acute angle opposite leg a, in degrees

Worked Example

Legs a = 3 cm and b = 4 cm.

  1. a² + b² = 9 + 16 = 25
  2. c = √25 = 5.000 cm
  3. A = arctan(3 ÷ 4) = arctan(0.75) = 36.87°
  4. Area = 3 × 4 ÷ 2 = 6.00 cm²
  5. Perimeter = 3 + 4 + 5 = 12.000 cm

Limits and common mistakes

  • The formula only works for a right triangle (one 90° angle). For other triangles use the law of cosines.
  • Use one unit for both legs: mixing inches and centimetres gives a wrong answer. The area is in square units of that same unit (cm², in², ft²).
  • The hypotenuse is always the longest side; if you know the hypotenuse and one leg, the missing leg is √(c² − a²).
Formula used
in a | Leg a | cm | 3 | 0
in b | Leg b | cm | 4 | 0
out c | Hypotenuse c | cm | sqrt(a^2 + b^2) | 3
out ang | Angle opposite leg a | degrees | atand(a / b) | 2
out area | Area | cm² | a * b / 2 | 2
out per | Perimeter | cm | a + b + c | 3

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.