A 30-60-90 right triangle always has the same fixed side ratios — this finds all three sides from just one known value.
How it works
The short leg, long leg, and hypotenuse are always in the ratio 1 : √3 : 2 — entering any one of the short leg or hypotenuse gives all three sides directly.
What this does not include
This does not include general right triangles with arbitrary angles — for those, use this site’s hypotenuse calculator or right-triangle side-and-angle calculator instead.
How to use this calculator
- Choose what you know, then enter the value.
A worked example
A 30-60-90 triangle with a short leg of 5: long leg = 5 × √3 = 8.660254, hypotenuse = 5 × 2 = 10.
Given a hypotenuse of 10, the short leg is 5 and the long leg is 8.660254 — the reverse calculation.
What the variables mean
| Variable | Meaning |
|---|---|
| Short leg | The side opposite the 30° angle |
| Long leg | The side opposite the 60° angle, always short leg × √3 |
| Hypotenuse | The side opposite the 90° angle, always short leg × 2 |
Edge cases worth knowing
The three sides always follow a fixed 1 : √3 : 2 ratio — knowing just one side length is enough to determine the other two exactly, unlike a general triangle where more information is needed.
A short leg of zero collapses the whole triangle, so the calculator declines to show a result — there’s no meaningful shape left to describe.
Frequently asked questions
Why is the long leg always √3 times the short leg?
It follows directly from the Pythagorean theorem applied to this specific angle combination — a 30-60-90 triangle is exactly half of an equilateral triangle cut through its height.
Where does a 30-60-90 triangle show up in practice?
Common angles in construction, engineering, and geometry problems, particularly anything involving equilateral triangles cut in half.
Is the hypotenuse always exactly twice the short leg?
Yes — that fixed 2:1 ratio is one of the defining properties of this specific special triangle.
