Give any two measurements and the rest follows: the third side, both acute angles, the area, the perimeter and the circles that fit inside and around it.
Result
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A right triangle is fully determined by two facts, because the right angle is the third one you already have. Two sides fix it through Pythagoras. A side and an angle fix it through trigonometry. Nothing else is needed, and nothing else is free to vary.
Pythagoras handles the sides: the square on the hypotenuse equals the sum of the squares on the other two. It is a statement about areas, which is why the squares appear, and it holds only when one angle is exactly ninety degrees.
When an angle is involved, the three ratios take over. Sine relates the opposite side to the hypotenuse, cosine the adjacent side to the hypotenuse, and tangent one short side to the other. Every one of them is a fixed number for a given angle, no matter how large the triangle — which is precisely what makes them useful.
a² + b² = c²
sin A = a ÷ c cos A = b ÷ c tan A = a ÷ b
A = a × b ÷ 2 h = a × b ÷ c
The altitude is the perpendicular from the right angle down to the hypotenuse.
Sides of 3 and 4 give a hypotenuse of exactly 5 — the oldest trick in building, used to square a corner without an instrument.
The angles are about 36.87° and 53.13°. Any multiple works too, which is why builders reach for 30-40-50 or 60-80-100 on site.
A ladder 5 m long reaching a window 4 m up stands 3 m from the wall, at about 53° to the ground.
Ladder guidance usually asks for something near 75°, so this one is set too shallow — the calculator makes that visible immediately.
A rise of 2 m at a 30° pitch gives a horizontal run of about 3.46 m and a rafter of 4 m.
The rafter is the hypotenuse: always longer than either the rise or the run, and the length you actually have to buy.
Always the one opposite the right angle, and always the longest of the three.
If a calculation gives you a hypotenuse shorter than one of the other sides, two values have been swapped somewhere.
Because it makes a perfect right angle out of a tape measure alone. Mark 3 units along one line, 4 along the other, and adjust until the diagonal reads exactly 5.
It scales freely, so bigger multiples give a more accurate corner over a longer wall.
No — the angle field accepts degrees, radians or gradians, and converts internally.
Degrees are the usual choice for construction; radians are what the underlying trigonometry actually uses.
It is the shortest distance from the right-angle corner to the opposite side, and it appears constantly in structural and optical problems.
It also splits the triangle into two smaller ones, each similar to the original — a fact worth knowing when a problem seems to be missing a measurement.
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