How this instrument works
Sine and cosine turn a single angle into both legs of a right triangle once the hypotenuse is known. The opposite leg equals the hypotenuse times the sine of the angle, and the adjacent leg equals the hypotenuse times the cosine of that same angle — two ratios built into every right triangle sharing that angle, no matter how large or small the triangle actually is. Fix the angle at 30° and the proportions never shift; only the hypotenuse's length scales the whole picture up or down.
This sheet is not the only right-triangle solver on the site, and the difference matters. A companion page starts from one leg plus the hypotenuse and works out the missing leg and both acute angles from there. This one begins somewhere else entirely: a directly measured angle, read off a protractor or inclinometer, paired with a hypotenuse you already know. That pairing suits a surveyor sighting a rooftop at a fixed distance, or anyone who trusts an angle reading more than a second length measurement.
Three worked cases make the pattern concrete. A 30° reading with a hypotenuse of 10 splits into a short opposite side and a longer adjacent side. Push the reading to 45° and the two sides become equal, since sine and cosine agree exactly at that midpoint — the signature of an isosceles right triangle. Push it further to 60° and the proportions flip, with the opposite side now the longer of the pair. None of this needs either leg known beforehand; the angle and hypotenuse alone pin down the whole shape.
- Pick a unit for the Known angle field — degrees, radians, or turns — and enter your measured value.
- Enter the Hypotenuse length in whatever unit you measured it in.
- Read Opposite side, the leg facing the angle you entered.
- Read Adjacent side, the leg lying along that same angle.
- Switch the angle unit at any time; both outputs recalculate immediately.
Worked example — 30°, hypotenuse 10
A rangefinder reads an angle of 30° up to a rooftop, and a laser gives a direct hypotenuse distance of 10 units. The opposite side is 10 × sin(30°) = 10 × 0.5 = 5.0 — the rooftop's height above the sighting point. The adjacent side is 10 × cos(30°) ≈ 10 × 0.866025 = 8.660254037844387 — the horizontal distance covered.
Change the reading to 45° and the hypotenuse to 10√2 ≈ 14.142136, and both sides land on exactly 10, since sine and cosine of 45° match — an isosceles right triangle falls out automatically. At 60° with a hypotenuse of 4, the opposite side comes to 4 × sin(60°) ≈ 3.4641016151377544 and the adjacent side to 4 × cos(60°) = 2.0 — fixed 30-60-90 proportions, reached this time from an angle and hypotenuse instead of a leg.
Questions
What's the formula for finding a leg from an angle and the hypotenuse?
Multiply the hypotenuse by sine for the opposite leg, or by cosine for the adjacent leg. With a 30° reading and a hypotenuse of 10, that's 10 × sin(30°) = 5 for the opposite leg and 10 × cos(30°) ≈ 8.66 for the adjacent leg — the SOH and CAH halves of SOH-CAH-TOA, applied directly.
Why start from an angle and the hypotenuse instead of a leg and the hypotenuse?
Because that's genuinely what's in hand sometimes — an inclinometer or protractor gives a reading directly, without needing a second length at all. Pair that reading with a known hypotenuse, such as a fixed cable or sighting distance, and both legs fall out immediately, no leg measurement required first.
What happens at exactly 45°?
The opposite and adjacent legs come out equal, since sine and cosine agree at 45° — each equals about 0.707107. A hypotenuse of 10√2 therefore splits into two legs of exactly 10 apiece, the defining mark of an isosceles right triangle.
Can the angle be entered in radians or turns instead of degrees?
Yes — switch the unit on the Known angle field to radians or turns and enter your reading in that unit; the sheet converts internally before applying sine and cosine, so both leg outputs stay correct regardless of which unit gets used.
Does the hypotenuse always have to be the longest side?
Yes — in any right triangle the hypotenuse sits opposite the right angle and is necessarily longer than either leg, so the two computed legs will always come out shorter than whatever hypotenuse value gets entered.
What if the angle is very close to 0° or 90°?
Near 0°, the opposite leg shrinks toward zero while the adjacent leg approaches the full hypotenuse length; near 90°, the reverse happens. The shape grows increasingly flattened at either extreme, though it never fully collapses within the allowed input range.