How this instrument works
Given one leg and the hypotenuse of a right triangle, the entire shape is already determined. The other leg follows directly from the Pythagorean theorem, otherLeg = √(hyp² − leg²), and the angle opposite the known leg follows from basic right-triangle trigonometry, angle₁ = asin(leg ⁄ hyp) — the SOH step in SOH-CAH-TOA, run in reverse. The remaining acute angle is simply whatever's left of the 90° total: angle₂ = 90° − angle₁.
This page starts from a genuinely different pair of known measurements than a right-triangle solver built around two known LEGS — here, the hypotenuse is one of the two starting inputs instead, exactly the situation whenever the hypotenuse (often the easiest side to measure directly, such as a ladder's own length leaned against a wall) is known alongside just one leg.
Every right triangle solved this way satisfies a built-in consistency check worth knowing: the known leg can never exceed the hypotenuse, since the hypotenuse is always a right triangle's longest side by definition — entering a leg longer than the stated hypotenuse describes a geometrically impossible triangle.
- Enter the known leg length into the Known leg field.
- Enter the hypotenuse length into the Hypotenuse field.
- Read Other leg, Angle opposite known leg, and Angle opposite other leg: the sheet solves all three simultaneously.
Worked example — leg 3, hypotenuse 5
A right triangle has a known leg of 3 and a hypotenuse of 5. The other leg is √(25−9)=4, the angle opposite the known leg is asin(0.6)≈36.87°, and the remaining acute angle is 53.13° — the full triangle solved from just those two starting measurements, the familiar 3-4-5 right triangle recovered directly.
A leg of 5 with a hypotenuse of 5√2 solves to an isosceles right triangle: the other leg is also 5, and both acute angles are exactly 45° — a special case where the known leg happens to equal the missing one.
Questions
How do you solve a right triangle from a leg and the hypotenuse?
Find the other leg with the Pythagorean theorem, √(hyp²−leg²), and the angle opposite the known leg with inverse sine, asin(leg⁄hyp). The remaining acute angle is simply 90° minus that first angle.
Why start from a leg and the hypotenuse instead of two legs?
Because that's often exactly what's directly measurable — a ladder's own length against a wall, or a support cable's length against a pole, gives the hypotenuse and one leg directly, without the second leg ever needing separate measurement.
What if the leg is longer than the hypotenuse?
No real right triangle exists with those measurements — the hypotenuse is always the longest side of a right triangle by definition, so a stated leg exceeding it describes a geometrically impossible combination, and this calculator flags that input.
Do the two acute angles always sum to 90°?
Yes, always — a right triangle's three angles sum to 180° total, and one of them is already fixed at 90°, leaving exactly 90° to split between the remaining two acute angles.
What happens if the leg equals the hypotenuse?
That combination is geometrically impossible — the hypotenuse must always be strictly longer than either leg, so a leg exactly equal to the hypotenuse describes a degenerate, flattened triangle with no real height at all.