How this instrument works
Two legs are already enough to pin down a right triangle completely. The angle between the hypotenuse and the adjacent leg comes from the inverse tangent of one leg divided by the other, angle = atan(opposite ⁄ adjacent), while the hypotenuse comes straight from the Pythagorean theorem, hyp = √(opposite² + adjacent²). Neither calculation needs a separate angle or hypotenuse supplied up front — both legs alone carry all the information required.
Three separate calculators on this site solve a right triangle, each starting from a different known pair. One begins with a single leg and the hypotenuse. A sibling page begins with a directly measured angle and the hypotenuse. This one is the third option, starting from both legs directly — the situation whenever a carpenter, surveyor, or student has already measured the rise and the run and wants the slope angle and the diagonal length that connects them.
Two edge cases show the formula's reach. Shrink one leg to zero and the angle collapses to 0°, since there's no rise left to measure against the run — a flattened, degenerate figure where the hypotenuse simply equals the surviving leg. Make the legs match instead and the angle locks at exactly 45°, the isosceles right triangle's own mark, with the hypotenuse stretching out to that leg length times the square root of two.
- Enter the Opposite leg measurement.
- Enter the Adjacent leg measurement in the same unit.
- Read Angle, reported between the hypotenuse and the adjacent leg.
- Read Hypotenuse, the diagonal connecting both legs.
- Switch the angle's output unit between degrees, radians, or turns as needed.
Worked example — legs 3 and 4
A right triangle has an opposite leg of 3 and an adjacent leg of 4. The angle between hypotenuse and adjacent leg is atan(3 ⁄ 4) ≈ 36.86989764584402°, and the hypotenuse is √(3² + 4²) = √(9 + 16) = √25 = 5.0 — the familiar 3-4-5 triangle, reached this time from its two legs rather than from a leg and the hypotenuse together.
Drop the opposite leg to 0 while keeping the adjacent leg at 5, and the angle flattens to exactly 0° with a hypotenuse of 5 — no rise at all, just the run. Make both legs match at 5 apiece instead, and the angle becomes exactly 45°, with a hypotenuse of 5√2 ≈ 7.0710678118654755, the isosceles right triangle's defining proportions.
Questions
How do you find the angle from two known legs?
Divide the opposite leg by the adjacent leg and take the inverse tangent: angle = atan(opposite ⁄ adjacent). With legs of 3 and 4 that's atan(0.75) ≈ 36.87°, the angle sitting between the hypotenuse and the adjacent leg.
How is the hypotenuse found from two legs?
Square both legs, add the results, and take the square root — the Pythagorean theorem, hyp = √(opposite² + adjacent²). Legs of 3 and 4 give √(9+16) = √25 = 5 exactly, one of the cleanest whole-number right triangles there is.
What if one leg is zero?
The shape flattens completely: the angle becomes 0°, and the hypotenuse simply equals whichever leg remains. A zero opposite leg paired with an adjacent leg of 5 yields a hypotenuse of 5 and an angle of 0°, since there's no vertical rise to measure.
Why start from two legs instead of a leg and the hypotenuse?
Because both legs are frequently the two lengths already in hand — a rise and a run, a width and a height — before either the diagonal or the angle between them has been worked out. Feeding in both legs directly skips needing either of those other quantities first.
What angle comes out when both legs are equal?
Exactly 45°, every time, regardless of how large the matching legs are. Two legs of 5 apiece give a hypotenuse of 5√2 ≈ 7.07 and an angle of precisely 45° — the isosceles right triangle's fixed mark.
Which angle does this calculator report — at the base or at the top?
It reports the angle between the hypotenuse and the adjacent leg specifically. Swap which leg gets labeled adjacent versus opposite, and the complementary angle comes out instead, since a right triangle's two acute angles always sum to 90°.