Right-angled triangle calculator
Fill in any two boxes and leave the rest empty. There is no dropdown asking which two you know, because the boxes you filled in already said so.
This one runs on your own device. It is worked out in your browser, so there is nothing to send and nothing to log. Where a tool on this site does send something out, that page says so and names where it goes.
Why the square on the hypotenuse
In a right-angled triangle the square built on the
longest side has exactly the same area as the two squares on the other
sides put together: a² + b² = c². For sides of
3 and 4 that is 9 + 16 = 25, and the hypotenuse is the square root of 25,
which is 5.
That is why the answer so often comes out untidy. 3, 4, 5 works because 25 happens to be a square number; most triangles are not so obliging and the hypotenuse is an irrational number that no decimal will ever finish.
Sine, cosine and tangent are three ratios, not three mysteries
Each is one side divided by another. The sine of an angle is the side opposite it over the hypotenuse; the cosine is the side next to it over the hypotenuse; the tangent is opposite over adjacent. In the 3-4-5 triangle the angle at A has a sine of 3÷5 = 0.6, a cosine of 4÷5 = 0.8 and a tangent of 3÷4 = 0.75.
Everything this page does with angles is one of those three divisions, or one of them run backwards. The reading shows which, with your numbers in it.
Why two is enough, and three is too many
A right-angled triangle has six measurements — three sides and three angles — but one angle is already known to be 90 degrees and the other two must add up to the remaining 90. That leaves two free choices, so any two of the boxes above fix the triangle completely.
Give three and you are probably about to contradict yourself. This page uses the first two it finds and says which they were, rather than trying to reconcile figures that may not agree.
Where to go next
Two neighbours in the same arithmetic:
- percentage calculator — gradients and proportions
- unit converter — when the sides arrive in different units
What this does not do
Right angles only. A triangle with no right angle needs three measurements rather than two, and the law of sines can give two valid answers from the same figures — which is a genuinely different page rather than an extra box on this one.
There is only one angle box, and that is deliberate rather than an omission. Two angles fix the shape but not the size — a triangle can be any size at all with the same angles — so a pair of angles would never have been an answerable question. Since one of the two you give must be a length, and the second angle follows from the first, one box is all there is to offer.