Right Triangle Altitude Calculator

What is the altitude to the hypotenuse?

In a right triangle, the altitude from the right angle to the hypotenuse is a perpendicular segment that touches the hypotenuse. It splits the original right triangle into two smaller right triangles. The calculation is closely tied to the Pythagorean theorem, which OpenStax explains in its lesson on right triangles and the Pythagorean theorem.

This calculator uses the two legs of the right triangle because that is the most common student input. It returns the hypotenuse, area, altitude, and the two hypotenuse segments created by the altitude.

Formulas used

The altitude formula comes from comparing two area formulas for the same triangle. The area is ab/2 using the two legs, and it is also ch/2 using the hypotenuse as the base. Setting those equal gives h = ab/c.

Worked example

Why this topic appears in geometry

Right-triangle altitude problems show up in similarity, geometric mean, and proof questions. The altitude creates three similar right triangles, which is why relationships such as h² = mn appear in some textbooks. This page gives the numeric answer and explains the geometry behind it.

Common questions

• How do I find the altitude to the hypotenuse of a right triangle?
  If you know the two legs a and b, first find the hypotenuse c using c = √(a² + b²). Then use h = ab/c. This works because the area of the right triangle can be written as ab/2 using the legs or ch/2 using the hypotenuse and its altitude.
• Is the altitude the same as a leg of the triangle?
  No. The altitude to the hypotenuse is a new segment drawn from the right angle to the hypotenuse. It is inside the triangle and perpendicular to the hypotenuse. It is usually shorter than both legs, though its exact length depends on the triangle.
• What are the segments on the hypotenuse?
  When the altitude touches the hypotenuse, it splits the hypotenuse into two smaller segments. If the legs are a and b and the hypotenuse is c, the segments are a²/c and b²/c. These segment formulas come from similar-triangle relationships.
• Why does h = ab/c work?
  The triangle has one area, but you can compute it two ways. Using the legs, area = ab/2. Using the hypotenuse as the base, area = ch/2. Setting ab/2 = ch/2 and solving for h gives h = ab/c.
• Can I use this for non-right triangles?
  No. These specific formulas assume a right triangle. Non-right triangles still have altitudes, but the hypotenuse does not exist because the word hypotenuse only applies to the side opposite the right angle in a right triangle.
• What is the geometric mean relationship?
  The altitude to the hypotenuse is the geometric mean of the two hypotenuse segments: h² = mn. This relationship is often taught after students learn that the altitude creates two smaller triangles similar to the original right triangle.