How this instrument works
Take any right triangle and drop the altitude from its right angle straight down onto the hypotenuse. That single line does two things at once: it splits the hypotenuse into two segments, call them p and q, and it creates two smaller triangles, each similar to the original and to each other. That triple similarity produces a clean geometric-mean relationship — the altitude itself works out to the square root of p times q, altitude = √(p × q).
A separate tool on this site checks similarity between two entirely different right triangles by comparing each one's own leg ratio — feed it two separate shapes and it reports whether one is a scaled copy of the other. This page asks a different question altogether: given a single right triangle's own altitude and the two segments it creates on that one triangle's hypotenuse, what is the altitude's length? Nothing here compares two separate shapes; everything happens inside one triangle.
The special case worth knowing: when p and q come out equal, the altitude also equals that shared value, and the original triangle is an isosceles right triangle — the one shape where the altitude lands exactly at the hypotenuse's midpoint. Move away from that balance and the segments spread apart while the altitude, always the square root of their product, keeps landing somewhere between the smaller of the two and the larger.
- Enter Segment p, the hypotenuse piece nearer the shorter leg.
- Enter Segment q, the piece nearer the longer leg.
- Read Altitude to the hypotenuse — the length of the perpendicular dropped from the right angle.
- Set p and q equal to each other to see the isosceles special case, where the altitude matches both segments exactly.
Worked example — segments 4 and 9
Segment p = 4 and Segment q = 9. Multiply: 4 × 9 = 36, and altitude = √36 = 6 exactly — a clean whole number, since 36 happens to be a perfect square.
Set both segments to 1, and altitude = √(1 × 1) = 1 — the isosceles case, where the altitude matches each equal segment exactly. Try p = 2 and q = 8 instead: altitude = √16 = 4, another perfect square landing on a whole number even though the two segments themselves aren't equal this time.
Questions
What is the geometric-mean relationship in a right triangle?
When the altitude from a right triangle's own right angle lands on the hypotenuse, it splits that hypotenuse into two segments, p and q, and the altitude turns out to equal their geometric mean: √(p × q). With p = 4 and q = 9, that gives √36 = 6 exactly.
How is this different from the similarity checker elsewhere on this site?
That checker compares two entirely separate right triangles, testing whether their leg ratios match to confirm one is a scaled copy of the other. This page stays inside a single triangle: it takes the two segments one triangle's own altitude creates on its own hypotenuse and returns that altitude's length — no second triangle is ever involved.
Why does dropping the altitude create similar triangles at all?
The altitude splits the original right triangle into two smaller ones, and each smaller triangle shares an acute angle with the original plus the newly formed right angle at the altitude's foot — matching angles that make all three triangles similar to one another. That shared similarity is exactly what forces the altitude to equal √(p × q) rather than some other combination of p and q.
What happens when Segment p equals Segment q?
The altitude comes out equal to both of them, and the original triangle turns out to be an isosceles right triangle — the one case where the altitude lands precisely at the hypotenuse's midpoint. With p = q = 1, altitude = √(1 × 1) = 1.
Does this calculator also find the triangle's two legs?
No — it solves only for the altitude, given the two hypotenuse segments p and q. The same geometric-mean family extends further, giving each leg from its own adjacent segment and the full hypotenuse, but that's a separate relationship this page doesn't compute.
Does it matter which segment I call p and which I call q?
No — multiplication doesn't care about order, so p = 4, q = 9 and p = 9, q = 4 both return an altitude of 6. The labels only help you keep track of which segment sits next to which leg in your own sketch.