Hypotenuse Calculator

Find the hypotenuse or a missing leg of a right triangle using the Pythagorean theorem with step-by-step solutions.

Select Calculation Mode

Result

Hypotenuse (c)
5
units
Side a3
Side b4
Hypotenuse c5
Perimeter12
Area6

Step-by-Step Solution

1Formula: c = sqrt(a² + b²)
2c = sqrt(3² + 4²)
3c = sqrt(9 + 16) = sqrt(25)
4c = 5
c = sqrt(a² + b²) = sqrt(3² + 4²) = 5

Understanding the Pythagorean Theorem

The Pythagorean theorem is one of the most fundamental relationships in geometry. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed algebraically: a² + b² = c², where c is the hypotenuse and a and b are the legs.

This theorem is named after the ancient Greek mathematician Pythagoras, although evidence suggests it was known to Babylonian and Indian mathematicians centuries earlier. It forms the basis for distance calculations in coordinate geometry, trigonometry, and many areas of physics and engineering.

Formulas Used in This Calculator

Find Hypotenuse

Given both legs a and b, the hypotenuse c is calculated as:

c = sqrt(a² + b²)

Find Missing Leg

Given hypotenuse c and one leg, the missing leg is:

a = sqrt(c² - b²)

Triangle Area

The area of a right triangle from its two legs:

Area = (a x b) / 2

Triangle Perimeter

Sum of all three sides of the right triangle:

P = a + b + c

Common Pythagorean Triples

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the equation a² + b² = c². These are especially useful because they produce right triangles with whole-number side lengths.

  • 3, 4, 5 — The most basic and well-known triple. Any multiple (6-8-10, 9-12-15) also works.
  • 5, 12, 13 — Another common triple frequently seen in geometry problems.
  • 8, 15, 17 — Useful for larger right triangles.
  • 7, 24, 25 — Often appears in competition math and standardized tests.
  • 20, 21, 29 — An example of a triple with consecutive legs.

Practical Applications

The Pythagorean theorem has countless real-world uses. In construction, it ensures walls are square by checking if a 3-4-5 triangle forms a right angle. In navigation, it calculates straight-line distances. Engineers use it to find cable lengths, structural supports, and diagonal measurements. It also underpins the distance formula in coordinate geometry: d = sqrt((x2-x1)² + (y2-y1)²).

Tips for Accurate Calculations

  • Always ensure the hypotenuse is the longest side of the right triangle.
  • When finding a missing leg, c² - b² must be positive; otherwise, the inputs are invalid.
  • Use consistent units (all in cm, all in inches, etc.) for both legs.
  • For decimal results, round to an appropriate number of significant figures for your application.