Hypotenuse

The hypotenuse is the side of a right triangle that is opposite the right angle (90-degree angle). It is always the longest side of the triangle, because each side is longer than the side opposite its corresponding angle. The other two sides, which form the right angle, are called the legs or perpendicular sides.

The hypotenuse is fundamental to geometry and trigonometry. Its length can be calculated if both legs are known, or if one leg and one angle are known.

Pythagorean Theorem

If the legs are denoted a and b, and the hypotenuse c, then the relationship a² + b² = c² holds. This is known as the Pythagorean theorem. Therefore, the hypotenuse is the square root of the sum of the squares of the two legs. For example: if the legs are 3 and 4, the hypotenuse is 5, because 9 + 16 = 25.

Trigonometry

In a right triangle, the hypotenuse serves as the reference when defining sine and cosine. Sine of an angle is the opposite leg divided by the hypotenuse; cosine is the adjacent leg divided by the hypotenuse. Therefore, if an angle and one leg are known, the hypotenuse can be calculated.

Other Properties

The hypotenuse is the diameter of the circle circumscribed around a right triangle; the center of this circle lies at the midpoint of the hypotenuse, and the distances from this center to all three vertices are equal. Additionally, the perpendicular drawn from the right angle to the hypotenuse divides the triangle into two smaller right triangles that are similar to each other and to the original triangle.